I find quite a lot when trying to learn some new mathematics or in general brush up on something one of the main difficulties is finding good books for particular areas of study, so I figured it would be nice to get a comprehensive list of recommended book. I'll start with a few I've looked at:

General interest books
Proofs from the book by Ziegler and Aigner
Road to Reality: A Guide to the Laws of the Universe by Penrose

General Topology
Basic Topology by Armstrong
Some quite good notes here (http://www.math.cornell.edu/~hatcher/Top/Topdownloads.html ) also
I've heard Topology by Munkres is good

Complex Analysis
Visual Complex Analysis by Needham (This is the only one I've looked at but I'm sure there is a better one if someone is looking for all the formal proofs and things, maybe try Real and Complex analysis by Rudin (not for the faint hearted though))

Algebra
Algebra by Birkoff and Maclane - This is basically an introduction to algebra but mostly from the point of veiw of category theory.
Galois Theory by Stewert is not too bad an introduction

Number theory
Elementary Number Theory by Jones and Jones seems quite good, although is a little basic, I'm yet to check it out but 'An Introduction to the Theory of Numbers' by Hardy and Wright is quite a classic and a little more advanced.

There are probably a few missing but it would be nice for everyone to contribute if they've read any really nice books.

I'll just go for the classic list of my favourite books that you haven't mentioned yet:

"Algebraic Geometry" by Hartshorne
"Algebraic Topology" by Hatcher
"A Comprehensive Introduction to Differential Geometry" by Spivak
"Characteristic Classes" by Milnor and Stasheff
"Categories for the Working Mathematician" by MacLane
"Lie Groups" by Bump
"Representation Theory, a First Course" by Fulton and Harris

All of these are truly excellent books on the subjects written in the titles.

Here are some more really good ones:

"Geometry of Differential forms" by Morita
Lots of really good material about differential geometry and differential forms in differential geometry; has nice descriptions of how to construct characteristic classes using curvature.

"Principles of Algebraic Geometry" by Griffiths and Harris
Very nice introduction to algebraic geometry for complex manifolds, including topics such as Riemann-Roch and many others.

"Differential Forms in Algebraic Topology" by Bott and Tu
A very nice book on the use of differential forms in algebraic topology, with interesting treatments of important theorems like Poincaré duality.

"Complex Functions: An Algebraic And Geometric Viewpoint" by Jones and Singerman
A nice introduction to Riemann surfaces and stuff like the Riemann Mapping Theorem and the modular group.

"Trees" by Serre
Very interesting book about trees with lots of examples (I liked the tree corresponding to the modular group for example!)

"An introduction to Knot Theory" by Lickorish
Very good book on Knot Theory by a leading expert on the subject.

"Riemann's Zeta Function" by Edwards
Excellent book about the Riemann Zeta function.

That's about all I can recall right now but I'll probably add a few more when I realise all that I forgot (or just when I read new books that turn out to be really good).

I think if you really want to get to grips with differential geometry though, you can't go wrong with the Warwick Business School's very own introduction: http://ideas.repec.org/p/wbs/wpaper/wp99-10.html

For some Carol Vorderman style brain training try: Berkley Problems in Mathematics, I just bought it in the Springer sale (still going on- can't find the catalog online though...). Its a text book full of sample problems/solutions for a PhD entry exam. Think 1,250 section C questions in a row with no sign of letting up, they're good quality and pretty varied, plus way more fun than SuDoKu.

A few that I like (which aren't in any of your posts) in no particular order:

The Arithmetic of Elliptic Curves
Advanced Topics in Elliptic Curves
Joe Silverman

These are both really really good and basically the standard texts on elliptic curves. A little advanced if you're trying to get into the subject I guess (as with most "standard" books), but you'll get quite a lot out of them.

Rational Points on Elliptic Curves
Joe Silverman and John Tate

This one is a pretty nice introduction, nowhere near as thorough as the previous two and quite informal but still very interesting and well written.

Introduction to Analytic Number Theory
Tom Apostol

The classic on analytic number theory and like most famous books that go "An introduction to..." (eg Hardy and Wright), it becomes fairly difficult after a while. That said, if you have a good enough foundation in analysis (real and complex) it shouldn't be too hard to follow and it proves lots of nice theorems, such as the PNT and Dirichlet's theorem.

Combinatorics
Peter Cameron

Awesome book on Combinatorics, has plenty of different topics and it's really interesting and well written. It has plenty of good exercises as well.

A Course in Arithmetic
Jean-Pierre Serre

Amazing little book on number theory. So much fascinating content packed into little over 100 pages, just really good in every aspect.

Basic Number Theory
Andre Weil

I haven't read this one but I had to mention it because it's got a wonderfully ironic title. It's a classic for class field theory and stuff like that though.

An Introduction to Number Theory
Everest and Ward

I quite like this book, it's a nice introduction to a lot of varied topics in number theory, from algebraic to analyic number theory through elliptic curves and with an excellent chapter on how all of these tie in beautifully.

Algebra
Serge Lang

A classic book on algebra, I haven't read much of it but it seems very complete and also very difficult.

Set Theory
Thomas Jech

Another classic, quite complete and just as terse. Still, it's enjoyable and great if you're serious about learning some set theory.

Elliptic Curves
Washington

I read some of this one for the elliptic curves course and it seems to be an all around very nice book. Again, not as detailed as Silverman but more accessible and still extremely interesting.

I'll stop here for now (mainly because I'm really hungry :P) but I'll add more later on. It's definetly a really good idea to have a thread like this. Perhaps I should look into making the books section more accessible and putting all of this there.

P.S. Thanks for reminding me about the Springer book sale, that's always a good time to get some maths books (even though the ones I'd like to get are usually still very expensive with 50% off :D).

P.P.S. That trees book looks amazing, I wanted to get it from the library but it's due back on the 1st of July...

By the springer sale do you mean: http://www.springer.com/sales?SGWID=3-40289-0-0-0 since I can't seem to find many good books there if I just browse through each section, unless I'm missing something?

Yeah that is what I was refering to. I didn't look through the whole list but I did spot a few interesting books: "Trees" (which Sam just bought) and "Galois Cohomology" by Serre, a few Bourbaki books, Van Der Waerden's "Algebra", etc. I'm not sure about most of them but there are probably a few other good ones in there as well.

There are far too few analysis books above! So, here is a list both of books I like and of books I'm not so fond of.

Big, must-have analysis tomes:
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Walter Rudin, Principles of Mathematical Analysis ("Baby Rudin") and Real and Complex Analysis ("Big Rudin")
Two absolute classics that everyone should own. Baby Rudin covers all of first and second year analysis in full generality (plus the basics of measure theory) in 342 pages. Big Rudin (which isn't much bigger) basically covers all the third-year analysis courses, but in such a way as everything is linked together and you can see the links between, say, measure theory, functional analysis, complex analysis (and even a little PDE theory, in terms of harmonic functions). It takes effort to read either, and both contain epic exercises. But very much worth it.

Elementary (i.e. first- and second- year) analysis:
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David Bressoud, A Radical Approach to Real Analysis and A Radical Approach to Lebesgue's Theory of Integration
Two books which have undoubtedly shaped my outlook on analysis. Instead of doing everything in the standard convergence-continuity-differentiation-integration order, they turn what we think of as normal on its head and do everything from an historical perspective. The former covers what you'd expect from Analysis I, II and III, and does a good job on historical context without sacrificing rigour; a must-read for anyone who already sort-of understands Analysis II but isn't so sure of why we're bothering. The latter is measure theory with as little metric spaces as possible, and while it doesn't sacrifice rigour, it does sacrifice generality; I prefer a slightly more abstract approach to measure theory, but I do appreciate Bressoud's historical angle.

Tom W. Körner, A Companion to Analysis: A Second First and a First Second Course in Analysis
Perfect for every second-year. It was my bedtime reading throughout first-term of second-year and it served me well; it assumes basically Analysis II and nothing more, acting as a fantastic bridge from the simplicity of the real line to the abstraction of metric spaces.

Peter Walker, Examples and Theorems in Analysis
Not a brilliant book, but it gets a mention for sticking its neck out: it is probably the only readable book that uses regulated functions and not the Riemann integral. Worth a look when you're doing Analysis III. [I know some of you will probably think why Warwick (and Oxford) bother swimming against the tide and refusing to do Riemann integration, but I really don't see the point in Riemann integration any more when we have Lebesgue integration. Plus, starting with regulated functions means that, apart from having to change "step function" to "simple function" and "uniform limit" to "pointwise limit", almost everything done for regulated functions goes through word-for-word to the Lebesgue integral. I think that's kinda neat.]

Measure theory:
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Marek Capinski and Ekkehard Kopp, Measure, Integral and Probability
A very well-written introduction to measure theory, and what Mario used to teach the third-year course from. Highly recommended. (Make sure you get the second edition, which has an extra chapter on the Radon-Nikodym theorem.) It manages to be very readable and yet entirely rigorous; what's more, it is unashamedly mathematically biased and yet it does justice to probability theory and even to applications such as finance. A must-read.

Paul R. Halmos, Measure Theory
Much-touted as a classic, but I'm not quite so sure. Yes, it's a good book, and its exposition is undoutedly clear and illuminating. But it's nearly 60 years old now, and measure theory has changed a lot in that time (if not in substance then at least in style) and the book shows its age somewhat. I haven't read it all, simply because I haven't needed to - there are other books that do the job. So don't write it off before you've tried it.

Russell A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock
Measure theory on steroids. A very clear and enjoyable account of Lebesgue integration, before launching into even more general integrals (yes, they do exist!), emphasising the concepts of absolute continuity and bounded variation. I haven't read much of it yet, but lines such as "It is not difficult (and not very much fun)..." make it a very good read.

Partial differential equation theory (yes, it is analysis!):
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Lawrence C. Evans, Partial Differential Equations
The bible. Probably the best textbook on PDEs ever written. It contains a wide variety of techniques for a wide variety of equations (there being a much richer variety of PDEs than ODEs). It is divided into three parts: classical methods, which usually involve finding an explicit solution formula; Sobolev space methods, which allow us to weaken the notion of "solution" to enable us to solve many more equations and give us a much more general theory for linear equations; and finally non-linear equations, which often arise from, and sometimes can be solved by, the calculus of variations. I simply would not have got through this year without it. (Note that it is a textbook: it is readable, and it does occasionally sacrifice some sharpness of theorems to achieve that.)

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order
A somewhat specialist but very good book on probably the most important kind of PDEs. I haven't read much of it yet, but I really like it, and I know Mario Micallef also really likes it. It tends to be the book that gets cited for the proof of this and that, it's a fantastic reference book. But really rather advanced.

Right, that'll do for analysis...

Algebra
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My favourite books for elementary (first- and second-year) algebra are:
John B. Fraleigh, A First Course in Abstract Algebra and
I. N. Herstein, Topics in Algebra

I do own P. M. Cohn, Classic Algebra, but I'm not fussed on it and it's hard to get into. Michael Artin, Algebra is one I ought to read properly at some point. But I don't like Niels Lauritzen: Concrete Abstract Algebra, which is what Dmitrii recommended for Algebra II in my year.

For group theory in particular I can recommend:
Geoff Smith and Olga Tabachnikova, Topics in Group Theory and
John F. Humphreys, A Course in Group Theory

For linear algebra I really like
Evar Nering, Linear Algebra and Matrix Theory
but it's out of print and mildly hard to get hold of (it's not in the library, which is annoying). If you can find it I recommend it, it's where I got some of the examples for the Algebra I revision guide.

For algebra from a geometric point of view (or perhaps geometry from an algebraic point of view?) I highly recommend
Miles Reid and Balazs Szendroi, Geometry and Topology
Very readable once you understand the basics of linear algebra.

And finally...
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Steven G. Krantz, Mathematical Apocrypha and Mathematical Apocrypha Redux
Two hilarious books (the second being the sequel, not a cut-down version) containing more anecdotes than I can begin to relate here; such as:

• Paul Halmos, claiming that he had just written the last word in measure theory, was challenged "Oh? And what is the last word in measure theory?" After going back to his office to check his just-finished book, he related that the last word in measure theory is "X".
• Some students at UCLA got fed up with how hard the qualifying exams were and wrote their own satirical version containing such gems as "State and prove the Prime Number Theorem, noting carefully each time that you use the commutativity of the reals."
• Stefan Bergman, stuck in France at the end of WWII with no papers, convinced a mayor to give him a piece of paper saying "This is to certify that Mr. Stefan Bergman has no papers". This was, of course, a paper, and with it he obtained the rations card he had hitherto been denied.
• J. D. Stein once published a book entitled "Sex, Crime and Functional Analysis: Part I: Functional Analysis".
• Late in life, Hilbert got confused reading a book and went next door to ask "What is a Hilbert space?"

Must-read books for every mathematician.

Hmm... I'm sure I've left something out... oh well, that'll do for now.

I am certainly not as versed in the mathematical scripture as many of the other contributors, but can suggest a few books...

For those interested in Combinatorics, the recommended text for the second year module is in fact an excellent read. Simply titled Combinatorics And Graph Theory (by Harris, Hirst, and Mossinghoff), the book starts at the beginning of combinatorics/graph theory and covers all the real basics. It goes on to topics including Colourings and Ramsey Theory. This stuff is all included in the second year module, but the book's final chapter extends this material in considering how we can study the combinatorial properties of infinite sets. In particular it emphasises the relations between combinatorics and set theory. Perhaps the greatest asset of this book is that it is really very humorous, which makes it all the more pleasant to read.

I'm currently also trudging my way through Enumerative Combinatorics, Volume I (by Stanley): this is a much harder book to read, but is certainly quite enjoyable for those who like to know how many derangements there are on an -set. Of course there is much more in this book than that. So far a very nice read, though the glancing at the exercises (where a difficulty level of 3 means that graduates may not be able to do the problem, and level 5 means unsolved question) may put some off.

Whilst on the topic of combinatorics... If anybody is interested in taking a third/fourth year reading module in this, I'm happy to try and set one up: send me a message if so.

Oh, and on a lighter note: I believe that Professor Stewart's Cabinet of Mathematical Curiosities is a very fun and interesting little book... though this one is not really about combinatorics... I'm sure something could be said about permuting the chapters or something though... hmm...

Yeah, I forgot to mention Rudin and Stanley in mine, I do really love those as well. For that matter, here are two more combinatorics books that I like which you can get for free on the internet (legally, you can obviously get most of these otherwise :P): Reinhard Diestel's Graph Theory and Herb Wilf's Generatingfunctionology. Another quite good introduction to combinatorics is Wilson and Van Lint's A Course in Combinatorics.

I did mention Rudin previously but another book I've recently come across which seems to have a lot of wonderful stuff in it (elliptic curves, fourier transform, theta functions, prime number theorem etc) is Complex Analysis by Stein (Princeton lectures in analysis).

I'm not a great fan of the Princeton Lectures in Analysis series, if only because the book on measure theory was the book that Carlos lectured MA359 Measure Theory from in my third year and it was terrible, and it also defined the support of a function differently to every other book on the planet (it just set rather than its closure, which is MUCH more standard). But to each his own.

For those aspiring logicians out there, I recommend Shawn Hedman's A First Course in Logic . I've been hogging a library copy for a while, and it has been jolly good fun.

I am also a big fan of Jech, but for the beginning set theorist, the more introductory and light, An Introduction to Set Theory by the same author (with Hrbáček) might be a better starting point.

For Lie algebras, myself and Colin have been finding Erdman and Wildon's book useful, and very well suited as a companion to the course lectured by Inna.

On a slightly less mathematical note, those with a passing interest in philosophical cosmology could do worse than pick up Space, Time and Spacetime by Sklar. I've not had time to get too deep into this one yet, so I'll let you know how it goes.

Another absolutely amazing book on algebra is David Eisenbud's 'Commutative Algebra with a view Toward Algebraic Geometry'. I've only just started looking at it and again not for the light hearted but I've heard it seriously recommended before looking at Algebraic Geometry by Hartshorne. So for anyone one day looking to classify some algebraic varieties...

Another absolutely amazing book which I've only recently had time to really look at is The Princeton Companion to Mathematics edited by Tim Gowers. Its got lots of wonderful and genuinely insightful articles in it by a host of famous names.

Another absolutely amazing book I've been reading is 'Euler:The Master of Us All' by William Dunham. A really wonderful account of his mathematical achievements (the content probably wouldn't be too hard for someone just out of highschool) with plenty of historical remarks too. Check it out!

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