Where can I find a book which explains the development of modern logic, e.g. Tarski, Frege, Peano, up untill Wittgenstein, Russel?

1$\begingroup$ Try Logicomix (logicomix.com/en). Electronic version: gen.lib.rus.ec/… $\endgroup$– Dmitri PavlovJan 18, 2012 at 9:47

3$\begingroup$ Logicomix is fun, but you won’t learn any logic from it. $\endgroup$– Emil JeřábekJan 18, 2012 at 11:31

1$\begingroup$ A suggestion I haven't seen yet is the Handbook of the history of logic. It is a multivolume treatise and several of the volumes are relevant here. elsevier.com/wps/find/bookseriesdescription.cws_home/BS_S369/… $\endgroup$– Andrés E. CaicedoJan 18, 2012 at 20:10
5 Answers
The scope of the figures you mention (Tarski, Frege, Peano, Wittgenstein, Russell) makes it a little unclear exactly what you're after. For instance, From Frege to Goedel (as mentioned by Mahmud) is an excellent compilation of early texts in mathematical logic  you get e.g. Frege, Peano, Hilbert, Zermelo, Skolem, Herbrand, Goedel  with helpful introductions included, but the focus is on the primary texts, rather than giving a single, unified account of the development of logic. And its relative lack of a philosophical focus means there's nothing like Russell or Wittgenstein to be found. [N.B. Along similar lines to this work, the two volumes of From Kant to Hilbert offer a more wideranging (in terms of subject and chronology) crosssection of works in the foundations of mathematics; note, though, that mathematical logic per se is not the focus there.]
Not knowing your background, or your exact goal, I would tentatively recommend Benacerraf and Putnam's Philosophy of Mathematics: Selected Readings. It has a great selection of works by the likes of Frege, Russell, Hilbert, Brouwer, Goedel, Von Neumann, Quine, and so on (and Wittgenstein is mentioned aplenty). In total, you get a lot about the interplay between technical matters in mathematical logic, foundations of math, and also related issues of a more straightup philosophical nature (if you're into that sort of thing). It too doesn't give a single chronological narrative, but just skipping around the articles in that collection will give you a lot to chew on, and ultimately give you a better account of the development of modern logic than will primary sources (IMHO).
Although it's not exactly what you asked for, you might take a look at the book "Foundations of Mathematics" by William S. Hatcher. It's primarily about the various foundational systems themselves, but my recollection is that Hatcher includes a good deal of historical information.
I would suggest to start with "The Search for Mathematical Roots, 18701940" by I. GrattanGuiness, which has some chapters on Cantor, which you can skip if it really doesn't interest you, so that you begin with chapter 4, which is on Peirce and Frege. It has less on Tarski than you might want, in which case you can read the Feferman biography called "Alfred Tarski: Life and Logic". It may also have less on Wittgenstein than you want, but there are myriad supplemental materials there.
There was an FOM thread a while back mentioning a number of titles:
Look for messages with the subject "Book on the history of logic?". There are a couple more in the following month of the FOM archive.