Free, high quality mathematical writing online? I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find.  Many of these turn out to be somewhat unreadable because of writing quality, organization or presentation.
I recently found out that "The Elements of Statistical Learning' by Hastie, Tibshirani and Friedman was available free online: http://www-stat.stanford.edu/~tibs/ElemStatLearn/ .  It is a really well written book at a high technical level.  Moreover, this is the second edition which means the book has already gone through quite a few levels of editing.
I was quite amazed to see a resource like this available free online.
Now, my question is, are there more resources like this?  Are there free mathematics books that have it all: well-written, well-illustrated, properly typeset and so on?
Now, on the one hand, I have been saying 'book' but I am sure that good mathematical writing online is not limited to just books.  On the other hand, I definitely don't mean the typical journal article.  It's hard to come up with good criteria on this score, but I am talking about writing that is reasonably lengthy, addresses several topics and whose purpose is essentially pedagogical.
If so, I'd love to hear about them.  Please suggest just one resource per comment so we can vote them up and provide a link!
 A: Check out Allen Hatcher's online books (topological stuff).
A: Volumes 28 through 56 of the MSRI book series are available here:
http://www.msri.org/communications/books/index.html
A: Paul Garrett is quite the author:
http://www.math.umn.edu/~garrett/
He has a book on buildings and many vignettes about automorphic forms, L-functions, representation theory, .... He wrote a graduate algebra book while he taught the course, and promptly got it published.
http://www.math.umn.edu/~garrett/m/algebra/
A: Robert Ash is a professor who's in the habit of making his textbooks available online as well.
A: The two-volume relatively introductory work on operator algebras, Operator Algebras and Quantum Statistical Mechanics by O. Bratteli and D. Robinson is available at Bratteli's website.
A: MAA Writing Awards
A: I recommend Mel Hochster's notes. The notes for Math 614 and 615 form an introduction to commutative algebra, and 711 is on a different topic (tight closure, Henselization, etc.) every year. I think they're very easy to read.
A: Stephen Boyd has some good books on his Stanford home page:
http://www.stanford.edu/~boyd/books.html ... especially the one on convex optimization is very good.
A: The wonderful webpage of Milne has books/lectures notes on a wide variety of topics, including Algebraic geometry, Etale cohomology, Class field theory, ...
A: A few more recommendations, apparently less well-known:
Saha's "Principles of Data Analysis" (you seem to have an interest in that field)
Noam Elkies' Lecture Notes (e.g this one on Analytic Number Theory) are like small books.
"Algorithmic Game Theory" by Nisan, Roughgarden, Tardos and Vazirani.
A: IF you want to see free academic video courses from leading universities, just go to 
http://www.academicearth.org/
A: Information Theory, Inference, and Learning Algorithms by David MacKay of the Cavendish Laboratory.
A: A great hidden gem is Shlomo Sternberg's page of online books:
http://www.math.harvard.edu/~shlomo/
Also, Curt McMullen has some notes at the bottom of this page
http://www.math.harvard.edu/~ctm/papers/index.html
which are good, but less formal. He also has other notes on his website not listed there; just look at his list of past courses and follow the links.
A: I'm glad someone mentioned Keith Conrad's notes, as they are excellent.
I would also like to point people towards Tom Weston's webpage. He has expository papers at http://www.math.umass.edu/~weston/ep.html on several topics, including cobordism theory and spectral sequences.
He also has some course notes at http://www.math.umass.edu/~weston/cn.html, including truly excellent book-length notes on introductory algebraic number theory, as well as several dozen illuminating pages on local fields and ideles.
A: In nlab we keep a list of main links of archives and free book collections in our main areas of interests (we were intentionally selective there): 
For top level directory for math resources see http://ncatlab.org/nlab/show/math+resources, from where you can go to archives, individual author collections, blogs and institutions.
A: Everybody probably knows about this already, but Allen Hatcher's textbook on Algebraic Topology is excellent - clear, well-written, neatly typeset.  It takes the student from basic concepts like homotopy equivalence all the way through to things like higher homotopy groups, obstruction theory and representability.
(His partially-written books on K-Theory and Spectral Sequences are also worth a look.)
A: I'm sort of surprised nobody has mentioned Terry Tao's blog yet. I think it definitely belongs in this list. 
A: Check out Jean-Pierre Demailly's books on analytic algebraic geometry http://www-fourier.ujf-grenoble.fr/~demailly/books.html.
Here you go the AMS book online webpage  http://www.ams.org/online_bks/online_subject.html .
I should also mention the AMS online book webpage collection
http://www.ams.org/online_bks/online-books-web.html.
A: You can find here large pool of ebooks in every brach of Math. 
A: The second edition of generatingfunctionology by Herbert Wilf is freely available online and is one of my favorite math books ever. It's one of the books that made me fall in love with combinatorics (the other being the Bollobas Graph Theory book).
A: "Linear Algebra" by Jim Hefferon has been online for a while and it was what I used to teach myself linear algebra. It's very well written with tons of great practice problems and interesting asides. It is a little less advanced than any of the other books listed so far, but it's still a great read. Plus, it's open source (You can download the LaTeX for the book from the website).
A: And another great source of the lecture notes and stuff is the MIT OpenCourseWare, in particular the math section.  
A: A draft of Albert Marden's Outer circles: an introduction to hyperbolic 3-manifolds is online, on his website:
http://www.math.umn.edu/~am/book/outercircles.pdf
Edward Nelson's Radically elementary probability theory is also online, on his website:
http://www.math.princeton.edu/~nelson/books/rept.pdf
A: The Stacks Project
http://math.columbia.edu/algebraic_geometry/stacks-git/
If I ever wonder about something, I can pretty much count on it to be in there. Remember to use a PDF viewer with hyperlinks and back/next buttons.
A: Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths is available through MSRI (and is sadly out of print at the moment).
http://library.msri.org/books/masterlist.html
A: Richard P. Stanley's Enumerative Combinatorics, volume 1, second edition is available
at http://www-math.mit.edu/~rstan/ec/ec1/ .
A: John Baez's stuff is a fantastic resource for learning about - well, whatever John Baez is interested in, but fortunately that's a lot of interesting stuff.  Scroll down for a link to TWF as well as his expository articles.
A: The wonderful book "A=B", by Marko Petkovsek, Herbert Wilf and Doron Zeilberger, is available freely online, thanks to their publisher AK Peters.
If you've ever wondered how to prove identities for q-multinomials and friends, well, the summary of this book is that computers now know how to do it, and you shouldn't bother anymore.
A: I hope it's not too rude to double-post, but as far as high-quality books go, Fulton's Algebraic Curves was also recently made available online.
A: Many know Hatchers Book, but few know the nice Concise Course in Algebraic Topology by J.P.May, which discusses, aside the standard stuff, Groupoids, Higher Homotopy and all that in a very brief and modern fashion. I think this is the book to read (for free) after/between Hatchers book.
There is also a big literature overview included, at the end of the book.
May has written much more (just look at his homepage), and I didn't read all of it. But what I read, I liked.
A: I was hoping that someone had posted Keith Conrad's expository stuff. Twice this week I've searched for an example in algebraic number theory (it is somewhat surprising how few of these there are in the books I own) and found the perfect answer on that page. The papers are remarkable for their high number of carefully chosen examples, just enough of which are worked out for the reader.
A: 'The Elements of Statistical Learning' by Hastie, Tibshirani and Friedman
http://www-stat.stanford.edu/~tibs/ElemStatLearn/
A: Len Evens has a couple of online textbooks: a text on abstract algebra, and a linear algebra text.
A: Martin J Osborne and Ariel Rubinstein's book "A course in Game theory" is available here.
A: The Caltechbook service at Caltech offers a number of math books for free here, including some very good (IMHO) books by Jerry Marsden et al.
A: Sergei Winitzki has an interesting-looking book on the coordinate-free approach to linear algebra online.
A: Jerome Keisler's Elementary Calculus.  This book uses infinitesimals explicitly, and also in a logically rigorous way, without getting too advanced for first-year undergraduates.
Later edit: http://math.wisc.edu/~keisler/calc.html
A: Gerald Teschl books


*

*Textbook Ordinary Differential Equations and Dynamical Systems

*Textbook Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators


can be found at http://www.mat.univie.ac.at/~gerald/
A: Within the framework of the project retro.seals.ch, scientific journals are retrodigitized and made available via internet. The project contains the following mathematical journals:


*

*Commentarii Mathematici Helvetici

*Elemente der Mathematik

*Elemente der Mathematik (Beihefte zur Zeitschrift)

*L'Enseignement Mathématique
A: And just because I like the book so much, Flajolet and Sedgewick's Analytic Combinatorics is available online and is a great resource for learning about asymptotic analysis in combinatorics.  The first half is also a great introduction to various techniques for writing down generating functions.
A: Not really pointing to a book, but I'd like to let you know I'm soon (within a month or so) launching a site dedicated to this. It is now almost finished. It is going to be a place where people can add mathematical resources, vote on them, add reviews, see other people's favorites and so on. Books will be categorized by language, level, topics, status (draft, lecture notes, books) and so on. I hope I will be able to "advertise" it trough mathoverflow: as with many "social" sites, the more people join, the more interesting it will become.
EDIT: The site is now online. It's still young, but I hope it will improve with time; I certainly have to add some features, but I decided it was time to launch and see if people actually find it useful. You can find it here
A: http://mathunion.org/ICM/ has almost all volumes of ICM talks online
A: Abstract and concrete categories: The joy of cats by Jiri Adamek, Horst Herrlich and George Strecker, is a nice book for learning category theory.  It went out of print, so the authors made it available online for free.
A: Diestel's Graph Theory is probably not as canonical as Hatcher's textbook, but it's a very commonly used textbook for graduate courses in the subject, and it's a similarly broad basic reference.
A: I can't believe nobody mentioned :
NUMDAM and Göttinger Digitalisierungszentrum, where you'll find digitized versions of mathematical texts... monographies and articles which made mathematical history, but sometimes still count as important references!
A: http://www.cs.utk.edu/~dongarra/etemplates/book.html
A: Roland Speicher has some nice introductory material for Free Probability (mini course, survey articles etc.) All available at http://www.mast.queensu.ca/~speicher/survey.html
A: Jim Pitman's Combinatorial Stochastic Processes.
Later edit: http://works.bepress.com/jim_pitman/1
A: Book by David Levin, Yuval Peres and Elizabeth Wilmor on Markov chain theory and mixing times. http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf. It quickly takes someone with basic knowledge in probability and linear algebra into the heart of current research.
A: The website of the Leibniz award offers a free online collection: link, and there is the preprint server of the IHES.
A: The Comptes Rendus de l'Académie des Sciences are available online.
A: As with so many things, "There's a reddit for that":  http://www.reddit.com/r/mathbooks.  It's a mixed bag—much like searching for math books in a non-specialist bookstore, one gets the elementary mixed up with the sophisticated—but there are some gems there.
A: Basic Concepts of Enriched Category Theory by G.M. Kelly was originally published by Cambridge University Press in 1982 but is now available online:
http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html
My understanding is that it is the canonical reference for enriched category theory (and was written by the pioneer of the field).
A: http://www.oocities.com/alex_stef/mylist.html
