Old books still used It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like to get a quantitative result. So what are some "old" books that are still used?
Coming from (algebraic) topology, the first things which come to my mind are the works by Milnor. Frequently used (also as a topic for seminars) are his Characteristic Classes (1974, but based on lectures from 1957), his Morse Theory (1963) and other books and articles by him from the mid sixties. 
An older book, which is sometimes used, is Steenrod's The Topology of Fibre Bundles from 1951, but this feels a bit dated already. Books older than that in topology are usually only read for historical reasons.
As I have only very limited experience in other fields (except, perhaps, in algebraic geometry), my question is:

What are the oldest books regularly used in your field (and which don't feel "outdated")?

 A: Montgomery and Zippin "Topological Transformation Groups" (originally published in 1955) is still the only book to cover the relevant results on topological characterization of Lie groups in full generality (including Lie group actions). I am not sure if this belongs to algebra or topology area-wise, but it is used in my area, geometric group theory. 
For pedagogical purposes, I still use "What Is Mathematics?" by Courant and Robbins (originally published in 1941) and "Geometry and Imagination" (1932) by Hilbert and Kohn-Vossen, when a high school student or an undergraduate asks me for suggestions. 
My personal definition of an "old book" is the same as Lee Mosher's, so I do not include here Chapters 4-6 of Bourbaki's "Lie groups and Lie algebras" (1968) which I use as a working tool. 
A: "Differential and integral calculus" (Russian) by G. M. Fichtenholz was first published in 1948. Recently (in 2009) its $9^{th}$ edition was published and this book is still used as the main calculus textbook at some universities.
A: In metric geometry Busemann's "The Geometry of Geodesics" (1955) is still wonderful reading. This book is now published by Dover. 
A: I would like to add the nine volumes of the "Treatise on Analysis" of Jean Dieudonné (in French, "Éléments d'Analyse") which is quite thorough with beautiful exercises (unfortunately some of them contain errors or wrong hints) and give a broad view of contemporary aspects of Analysis, still useful nowadays especially the ninth & last volume (they were published in the 70s and  80s I think). Written with a flavor of Bourbaki, it gives the right level of generality (not too much, usually using only locally compact metrizable groups) and the numerous exercises really help to master maim results and methods of proof. 
A: Tate's thesis, Fourier analysis in number fields, and Hecke's zeta-functions, is from 1950 and is certainly still considered a primary on the subject (in addition to being the original resource).
A: Barry Simon and Michael Reed's classic volume on Functional Analysis (1981) is my one of my favorites.
Ayoub, "An Introduction to the Analytic Theory of Numbers," (1963) is out of print but one of the best books on the subject.
A: Many systematic introductions to the foundations of the edifice of Differential Geometry appeared in the sixties, and they are useful references even today. Some of them are:


*

*Lang, Introduction to Differentiable Manifolds, 1962;

*Helgason, Differential Geometry and Symmetric Spaces, 1962;

*Kobayashi, Nomizu, Foundations of Differential Geometry, 1st Vol 1963, 2nd Vol 1969;

*Sternberg, Lectures on Differential Geometry, 1964;

*Bishop, Crittenden, Geometry of Manifolds, 1964;

A: Two old books by David Mumford are not mentioned above (unless I am wrong):
1) Introduction to Algebraic Geometry (preliminary version of first 3 Chapters)
 (published and distributed by Harvard math. dpt., bound in red !, and containing 444 pages.)

  At that time (around end of 1960's ), this book was the unique good way to be introduced to theory of schemes . The EGA's were not helpful.
  In 1988, it became "The Red Book of Varieties and Schemes "(Springer). He is still excellent for learning ,and teaching , schemes.

2) The classical and fundamental:
      Geometric Invariant Theory (Springer, 1965),
  It has two enlarged editions : 1982, 1994.

A: 
H.S. Hall and S.R. Knight, Higher Algebra

First edition 1891 (or so), recent edition 2001 (for example). Subtitled a Sequel to Elementary Algebra for Schools, and so betrays the fact it's not really like Lurie's book of the same title.
A: Meet the Rudins: Baby Rudin (first published in 1953), Papa Rudin (whose oldest copyright I've been able to find dates back to 1966) and Grandaddy Rudin (1973 is the oldest reference I've found).
A: Hardy "Divergent series" (1949)
Naimark "Normed rings" (1968)
Maurin "Methods of Hilbert spaces" (1959)
Hille & Phillips "Functional analysis and semigroups" (1957)
A: Emil Artin's Geometric Algebra (Interscience, 1957) is definitely immortal.
A: EGA and SGA, both from the 1960s and 1970s, are very widely used in algebraic geometry. Hartshorne's textbook (first published in 1977) is still the main choice for courses on the theory of schemes.
A: "Introduction to commutative algebra" by Atiyah and MacDonald is from 1969. (I learnt commutative algebra from this book at the University of Oslo just a few years ago)
A: I'm amazed no one has mentioned Hardy and Wright's wonderful Introduction to the Theory of Numbers. It was first published in 1938 and is absolutely delightful.
The most recent (6th) edition includes a chapter on elliptic curves.
A: R. Engelking (1977). General Topology.
A: G.N. Watson's "A Treatise on the Theory of Bessel Functions" (1922), 
A: Keisler's "Calculus: An Approach Using Infinitesimals" is a very cool freshman calc book using NSA. It dates back to 1976, and is available for free online: http://www.math.wisc.edu/~keisler/calc.html . Although I'm not aware of anyone who's using Keisler in the classroom today, it's under a Creative Commons license, and there is a newer book by Guichard and Koblitz that incorporates a bunch of material from Keisler: http://www.whitman.edu/mathematics/multivariable/ . In the world of the digital commons, it's a little hard to define how old a book is. It's like asking how old a bacterium is. Bacteria are in some sense immortal. They just evolve.
Another wonderful old calc book that is still in print is Calculus Made Easy, by Silvanus Thompson, 1910.
I noticed that another answer to this question got heavily downvoted for referring to a book published in the 1980's. The question was: 'What are the oldest books regularly used in your field (and which don't feel "outdated")?' It didn't specify what "used" meant -- used in research, teaching, personal study, ...? The lower you get on the educational totem pole, the shorter the half-life of a book. Someone posted that they liked Disquisitiones Arithmeticae, but that doesn't mean it's being used for teaching number theory to undergrad math majors. For freshman calc, it is extremely unusual for anybody to use anything more than 5 years old. The community college where I teach has an explicit rule forbidding the use of books of more than about that age.
A: My first thought was Atiyah & Macdonald's 'An Introduction to Commutative Algebra' - which has already been mentioned - and 'anything by J.P. Serre' (that's old enough, of course!). It appears that not quite everything in this latter category has been mentioned; notably, 'Algebres de Lie Semi-simple Complexes', first published in 1966. There is also a later English translation, 'Complex Semisimple Lie Algebras' published in 1987. 
While not quite an introduction, I find myself referring back to this text often for its streamlined, beautiful exposition (a hallmark of Serre). It also has the best exposition of root systems I've encountered. 
Furthermore, another classic text on semisimple Lie algebras (J. Humphreys - 'Introduction to Lie Algebras & Representation Theory') is a 'fleshing out' of Serre's notes. Actually, Humphreys's textbook was first published in 1972 so might squeeze onto this list too?
A: Rudin's Principles of Mathematical 
Analysis, and Herstein's Topics in
Algebra if not heavily used, are the
ideal that many people strive to in 
teaching introductory analysis and abstract
algebra to undergraduates.
A: "Projective Geometry" by Coxeter (1963), "Finite Geometries" by Dembowski (1968) and "Projective Planes" by Hughes and Piper (1973), still serve as great textbooks for these topics. 
A: I'm surprised that no one has mentioned Emil Artin's beautiful monograph on the Gamma Function.  The economy and elegance are unsurpassed.
A: I think the absolute record (excluding Euclid) belongs to

E. T. Whittaker  G. H. Watson, A course of modern analysis.

According to the Jahrbuch database, the first edition was in 1915.
Moreover, this 1915 edition was an extended version of a 1902 book,
by Whittaker alone.
The last revision was in 1927.
The book is still in print, and widely used, not only by mathematicians
but by physicists and engineers.
Soon we will celebrate the centenary... It has 1056 citations on Mathscinet, by the way, and 8866 on the Google Scholar !
Perhaps this deserves a Guinnes book of records entry as a "textbook longest continuously in print".
And I suppose this is a record not only for math but for all sciences...
with the exception of Euclid and Ptolemy, of course:-)
If we include not only textbooks but research monographs there are plenty of other examples, even
older ones:

H. F. Baker, Abelian functions, was first published in 1897. Reprinted in 1995, and there is a new
  Russian translation.

Just out of curiosity, look at its current citation rate in Mathscinet:-)
They also reprinted

H. Schubert, Kalkül der abzählenden Geometrie, 1879, in 1979,

and again you can see from Mathscinet
that people are using this.
EDIT: A brief inspection of the most cited (and thus most used) books on Mathscinet shows that
a very large proportion of the most cited books are 30-40 years old. 
Which is easy to explain, by the way. Thus on my opinion, such books do not qualify for this list
(unless we want to make it infinite).
EDIT2: Today I accidentally found that 3 of the 4 copies of

G. H. Watson, Treatise on the theory of Bessel functions (first edition, 1922)

are checked out from my university library.
Mathscinet shows 1157 citations for the last 2 editions.
Another question is old papers which are still highly sited. A typical life span of a paper is much
smaller than that of a book. In the list of 100 most cited papers in 2011, I found only two papers
published before 1950 (One by Shannon and another by Leray).
A: Mac Lane's "Categories for the working mathematician" (1971).
A: Henri Cartan and Samuel Eilenberg published their Homological Algebra in 1956, although it was famously circulated for a long time before that. While that book more or less founded its subject, it is still quite useful. 
A: If computer science counts as math, then The Art of Computer Programming (first volume published 1968) would be a good example of a text that's still in wide use.
A: Cassels and Frohlich (editors) on class field theory is regularly reprinted.
A: I still think the exposition on elliptic functions in Jacobi's Fundamenta Nova (1829) is one of the best I've encountered if you are interested in the functional relationships.  A close second for me is Cayley's An elementary treatise on elliptic functions (1895), especially for the number of alternative proofs presented and the numerous relationships detailed.  Modern books tend to take the algebraic approach, which is obviously extremely important for understanding the true nature of the relationships here, but for those of us who study the field because of it's incidental use in combinatorics and generating functions, these older books are a wealth.
Also, I have a personal love of Gauss' Disquisitiones Arithmeticae (1798) because it introduced me to number theory at a young age in a way that was very natural and elegant.  Again, I appreciated it's approach to forms and related because it was all easily understandable with middle school algebra.
And finally more modern, for me Goldblatt's Topoi: The categorial analysis of logic (1979) is the best introduction to categories one could have, far better in my opinion than even Mac Lane's.  That it is also subversive propaganda for constructivism is also a huge bonus.
A: My field is dominated by older books, it seems. Gilmer's Multiplicative Ideal Theory came out in 1972 and it's nearly unmatched in the content it covers. We're currently using Kaplansky's Commutative Rings book for the Commutative Algebra course I'm taking at UCR; Atiyah and Macdonald's book is also considered a standard reference for those kinds of courses, and it came out in 1969. And, of course, you can't forget Bourbaki. I'm also partial to Zariski and Samuel's Commutative Algebra texts over other texts in the field, which came out in 1958 and 1961.
A: For ordinary differential equations there is:


*

*Theory of ordinary differential equations by Coddington and Levinson, McGraw-Hill Book Company, 1955


I'm not sure it is used in courses, but it is certainly still cited frequently, for example as a reference for Carathéodory type differential equations where the vector field is only integrable in time.
A: That depends if you speak of research books or advanced text book. In the second category, I should place


*

*Rudin's Real and complex analysis (1966),

*J.-P. Serre's Cours d'Arithmétique (1970) (hope you will forgive me),

*Lang's Algebra (1st Edt 1965).
In the first category, I see


*

*Kato's Perturbation theory of linear operators (1966),

*Courant & Hilbert's Methods of Mathematical Physics (1924),

*Courant & Friedrich's Supersonic Flow and Shock Waves (1948),

*V. I. Arnold's Mathematical methods of classical mechanics (1974).
A: How about:
G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities (1934, second edition 1952).
G. Pólya, G. Szegő, Problems and Theorems in Analysis (first German edition in 1925)
G. Szegő, Orthogonal Polynomials (1939)
A: Ahlfors' Complex Analysis. The 3rd edition is from 1978, but the book itself was written in the 50s. No other book comes close.
A: van der Waerden's Moderne Algebra was first published in 1930, I think. I use the book occasionally for my course, but am not sure which edition. 
A: Abramowitz and Stegun's Handbook of Mathematical Functions (1964) is still used.  As the August 2011 Notices article by Boisvert et al. says, 

The Handbook remains highly relevant today
  in spite of its age. In 2009, for example, the Web
  of Science records more than 2,000 citations to
  the Handbook. That is more than one published
  paper every five hours—quite remarkable!

In time it might be superseded by the NIST Handbook of Mathematical Functions (or its online version, the Digital Library of Mathematical Functions), but not yet.
A: I would like to mention about M. Postnikov's geometry series, Lectures on geometry which I always refer to when I need some coherent view inbetween geometry and analysis. 
Sometimes I may refer to Hopf and Alexanderoff's Topologie in order to gain some authority...
A: When I was an undergrad, at the turn of the millenium, I took a complex analysis class that used (an English translation of) Knopp's 1936 Funktionentheorie.
A: For combinatorial group theory, there are essentially two books which encompass most of the area before the advent of geometric group theory à la Gromov, and they still serve as the primary sources for a great deal of fascinating and intricate results.
These are both called Combinatorial Group Theory; the first is from 1966 by Magnus, Karrass, and Solitar, and the second is from 1977 by Lyndon and Schupp. 
A: The notes of the 1951-2 Artin-Tate seminar on class field theory (published in 1968, and re-issued in LaTeX form a few years ago with a new Introduction by Tate addressing subsequent developments) remains a fundamental reference in algebraic number theory, despite the abundant supply of more recent references on the subject. 
One reason is that it is the only reference outside the original research literature where one can find a complete treatment (with proofs) of certain key aspects of the theory such as the Grunwald-Wang phenomenon and Weil groups for class formations (especially the case of number fields, which lacks a bare-hands construction as for local fields and global function fields). Come to think of it, the general notion of Weil groups for class formations emerged from that seminar...The style of the proofs remains generally quite fresh.
A: The calculus and analysis texts of Michael Spivak and Tom Apostol come to my mind...at least they are still widely used in my land (Colombia) for undergraduate (serious) math courses.
A: I'm surprised that nobody has mentioned Serre's Corps locaux (Local Fields), his Cohomologie galoisienne (Galois cohomology) and his Représentations linéaires des groupes finis (Linear representations of finite groups).  
Other eternal texts in Number Theory include Artin's Algebraic numbers and algebraic functions and the Artin-Tate notes on Class field theory, Hasse's Zahlentheorie and his Klassenkörperbericht,  Hecke's Vorlesungen über die Theorie der Algebraischen Zahlen, Weyl's Algebraic Theory of Numbers, and Hilbert's Zahlbericht. 
A: Artin's Galois theory (1942) is still a classic. People in automata theory and finite semigroups still use Samuel Eilenberg's two volumes on the subject (1974). 
A: The standard, go to reference in geometric measure theory is still Federer's 1969 classic, Geometric Measure Theory. It is very rarely the first reference one uses since it is rather dense and there are other introductions and expositions, some of them very good.
A: Sz. Nagy-Foias: Harmonic Analysis of Operators in Hilbert Space (1970) is a still widely used and lively book (though there is a new updated edition in 2012).
T. Kato's Perturbation Theory book (1967) is also definitely in this category, though there is a 1980 second edition and a 1995 reprint.
Nelson Dunford, Jacob T. Schwartz: Linear Operators (1958,1963, 1971). I still take this book regularly into my hands. 
An other reference on differential equations is
J. L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems, 1972. It is still "the" reference.
A: Mathematical Analysis By  Zorich 
A: Most of the textbooks I use are quite new.  The old books are the exception.
The oldest book about mathematics I use is Hajós György: Bevezetés a geometriába, a textbook on elementary geometry (in the sense of Euclid).  The first edition is from 1950, I have a copy published in 1960.  (Edit: it seems there's a German translation.)
I'm also using Knuth's The Art of Computer Programming, does that count as old now?  The translation of the first volume is based on the second edition, of which the original was published in 1973.  (Edit: the above was accurate when I wrote this post.  Since then, I actually bought the third edition versions in original English, of which the first volume was published in 1997, so it no longer counts as an old book.)
A: In classical invariant theory, both "The Algebra of Invariants" by Grace and Young and "An introduction to the algebra of quantics" by Elliott are still much in use. The latest edition of Grace and Young is 1903 and of Elliott 1913.
A: Daniel Quillen's "Homotopical algebra", 1967.
A: 
O. Kellogg, Foundation of Potential Theory

The first edition of Kellogg's Foundation of Potential Theory was published in 1929.
Btw he was a student of David Hilbert.
A: P.A. MacMahon, Combinatory analysis, vols 1 and 2, Cambridge University Press, 1915–16.
A: If one needs to use tools from classical invariant theory or elimination theory then some books that come to mind are:


*

*Grace and Young "The Algebra of Invariants", 1903.

*Elliott "An Introduction to The Algebra of Quantics",
1913.

*Salmon "Lessons Introductory to The Modern Higher Algebra", 1876.

*Faa di Bruno "Théorie Des Formes Binaires", 1876.

*Faa di Bruno "Théorie Générale de l'Elimination", 1859.
and there are quite a few more.

For Salmon's book, the 4th edition of 1885 might be best. Indeed, as I learned from a paper by Macauley, it has a discussion (on p. 87) of Cayley's very general formula for the multivariate resultant as the determinant of a complex (see the book by Gelfand, Kapranov and Zelevinsky for a modern account and a reprint of Cayley's paper).
A: I used G. H. Hardy's A Course of Pure Mathematics (First edition 1908) when I taught undergraduate real analysis not so long ago. The care with which concepts are explained and the number of interesting problems and examples is, in my opinion, unmatched by newer books.
A: Probability Theory, by Feller.  Volumes I and II.  Oldies but goldies
A: Most good books in general topology are old. Here are some good topology books that I often refer to.
rings of continuous functions by Gillman and Jerison (1960)
Uniform Spaces by John Isbell (1964)
General Topology by Stephen Willard (1970)
Topology by James Dugundji (1966)
A: My own field, ergodic theory, is relatively young in that some concepts now regarded as fundamental -- Kolmogorov-Sinai entropy, for example -- were not fully formulated until around 1960. Nonetheless there are a couple of old books still in use and receiving citations:
E. Hopf, Ergodentheorie, 1937;
R. Halmos, Ergodic theory, 1957.
If the 1960s are sufficiently long ago to constitute "old" then there are many old references in probability which remain in heavy use, for example:
P. Billingsley, Convergence of probability measures, 1968;
L. Breiman, Probability, 1968;
and one of the classics of the field:
W. Feller, Introduction to probability theory and its applications, 1950.
Outside my own field, a much-cited number theory text which no-one has yet mentioned:
A. Khinchin, Continued fractions, 1936.
A: Spivak's five volume "Comprehensive Introduction to Differential Geometry" still gets a lot of use---particularly the first two volumes.
A: I've used Euclid's Elements
Halmos (several)
A: Gaston Darboux' magnum opus Leçons sur la Théorie générale des Surfaces et les Applications géométriques du Calcul infinitésimal (first edition 1890, I think; there is a second edition dating from around 1915) is still read by many differential geometers, and, as far as I know, it is still in print via the AMS Chelsea series.
A: I was just looking at HSM Coxeter's Regular Polytopes (1948) pretty recently, and it is still wonderful.
A: No one suggests Weyl's Classical Groups? It was first published in 1939. I don't know if researchers in representation theory and invariant theory value it nowadays, but it is still frequently cited in random matrix literature.
A: Mathematical Foundations of Statistical Mechanics by A. I. Khinchin.  The original edition in Russian was published in 1943 according to MathSciNet (MR Number=(17677)).
A: In numerical linear algebra, Gantmacher's The theory of matrices is still a widely read and cited text (see MathSciNet citations). The Russian original dates back to 1953 (thanks @Giuseppe), and the first English translation is from 1959.
A: Dickson's "History of the Theory of Numbers" is not only old (1919), but it reviews material which is even older. I found it extremely useful when calculating some family Gromov-Witten invariants in a recent paper with Jarek Kedra - while performing the arithmetic manipulations in Section 8, we would have been lost without the wealth of formulae in Dickson. I've no doubt the material appears elsewhere, but Dickson has a comprehensive and carefully historical approach.
A: My choice of books would be:


*

*Theory of Riemann-Zeta Function by E.C. Titchmarsh, (Oxford University Press)

*Theory of Functions by E.C. Titchmarsh (Oxford University Press, 1952).
A: Some volumes of Bourbaki, as Topological Vector spaces or Lie groups are still widely quoted.
A: N. G de Bruijn's Asymptotic methods in analysis is still the best reference for the topic.  The current 1981 Dover reprint edition is largely unchanged since the 1958 first edition.
A: Bonnesen and Fenchel, "Theorie Der Konvexen Korper" Springer, Berlin 1934 not available in English translation until 1987 although Eggleston's "Convexity" 1958 draws heavily on it.
A: An Elementary Treatise on Coordinate geometry of Three Dimensions. Macmillan 1910, reprinted upto 1950 or later. Apart from classical setting of analytical geometries contains early differential geometry with theory of Invariants. Found the book in pavement shop (Cost: 0.1 $ !)
A: I would add 
1/ Regarding logic and set theory:
(i) The consistency of the continuum hypothesis, Kurt Gödel, Princeton University Press, 1940
(ii) Set theory and the indeeandance of the continuum hypothesis, Paul Cohen, Walter Benjamin, 1966
2/ Regarding analysis:
Leçons d'analyse fonctionnelle, Frederic Riesz et Bela Nagy, Gauthier-Villars, 1955
Gérard Lang
A: Nathaniel Bowditch is generally regarded as a nineteenth century American mathematician . His American Practical Navigator has been in continous print since 1804.  It is still in use today judging from the comments on Amazon.  But perhaps this isn't what was meant by a mathematics book and perhaps navigation isn't to be considered applied mathematics.
