Suggested reading for renormalization (not only in QFT) What papers/books/reviews can you suggest to learn what Renormalization "really" is?
Standard QFT textbooks are usually computation-heavy and provide little physical insight in this regard - after my QFT course, I was left with the impression that Renormalization is just a technical, somewhat arbitrary trick (justified by experience) to get rid of divergences. However, the appearance of Renormalization in other fields of physics Renormalization Group approach in statistical physics etc.), where its necessity and effectiveness have, more or less, clear physical meaning, suggests a general concept beyond the mere "shut up and calculate" ad-hoc gadget it is served as in usual QFT courses. 
I'm especially interested in texts providing some unifying insight about renormalization in QFT, statistical physics or pure mathematics.
 A: There are several books that do this, from Renormalization: an introduction and Renormalization: an introduction to renormalization, the renormalization group, and the operator-product expansion, to Quantum field theory and critical phenomena and Renormalization methods: a guide for beginners; or the more classics Scaling and renormalization in statistical physics and Finite quantum electrodynamics: the causal approach.
I hope this helps…
A: A really good textbook on QFT, in a new and exciting approach is "Quantum Field Theory in a Nutshell", by Anthony Zee. It is a not so technical book in QFT, and with a deep insight in physics.
A: Other people provided lots of references, so I'll just state what I think about the subject.
If you are familiar with statistical physics part of the renormalization, you should already have a good grasp also on QFT renormalization (even if you don't yet know it!). The moral is the same here: divergences arise because our picture is only effective and, more generally, the theory doesn't account for all the realistic effects of the nature (like measurement).
The UV divergences appear because of infinite interaction energies and they point at the fact that the theory might be incomplete (i.e. it's just an approximation of some better underlying theory), so we are not really allowed to take infinite energy limit without somehow modifying our theory to accommodate for this.
What about IR divergence? Well, this limit again cannot be taken if you think about it a little but the reason is different from the UV case. IR limit lets you count with arbitrarily small energies. But is this really physical? What about our measurement? Are we really able to measure arbitrarily small energies? Well, of course not. But QFT doesn't know anything about our measuring devices, so it's not surprising that you again have to account for this by hand.
Another novel point of renormalization is a running coupling. And again, this arises precisely because we started to notice that coupling constants aren't really constants when one thinks more deeply about what it constitutes to measure something.
I think the whole point of renormalization can be stated pretty concisely: it arises because we realized how ignorant we were. Both of ignoring the fact that QFT is not the ultimate theory of everything and also ignoring the subject of measuring.
A: :-DDD I have written another pedagogical article about renormalization... Not, seriously, the article is blahblah and I had even forgotten I had written it. Probably it even contributed in the evaluation that expelled me from hep-th to hep-ph, who knows. But the list of references is useful and then it is an actual answer to the OP, so allow me to paste them here. It was at hep-th/0208180 
Note that some historical references (eg Borel 1928 and some comments in the body of the paper) are given only to suggest why people was not so afraid about divergences in 1930, it was even a hot topic in related areas.


*

*G.A Arteca, F.M. Fernandez, E.A. Castro, Large Order Perturbation Theory and 
Summation Methods in Quantum Mechanics, Lecture Notes In Chemistry, 53, Springer

*E. Borel, Lessons sur las series divergentes, 
ed. Gautier-Villars, 1928 (reprinted  editions Jacques Gabay)

*E. Brezin, J.C. LeGuillou, J. Zinn-Justin, Perturbation theory at 
large Order, I et II, Phys Rev D, v. 15, p. 1544 and Phys Rev D v. 15, p. 1558

*Ch. Brouder, Runge-Kutta methods and renormalization, European 
Physical Journal C v. 12, p. 521-534

*J.C. Butcher, An Algebraic Theory of Integration Methods Math. Comp. v. 26, p. 79

*A. Connes and D. Kreimer hep-th/9912092, as well as D. Kreimer 
             q-alg/9707029 and hep-th/0010059, and C-K hep-th/9904044

*F.J.Dyson, Phys Rev 85, p. 631 

*LY Chen, N. Goldenfeld, Y. Oono, Phys. Rev. E, v. 54 p. 376

*Feynman, Space-Time Approach to NR Quantum Mechanics Rev Mod  Phys 20, p. 367

*M. Gell-Mann and F.E. Low  Quantum Electrodinamics at Small Distances, 
Phys. Rev. v. 95, p. 1300

*G t'Hooft, Nucl Phys B 35, p. 167; G. t'Hooft and M.Veltman, Nucl. Phys. B44 p. 189  

*D.J. Broadhurst and D. Kreimer,  Renormalization tamed: 30 loop Pade Borel 
Resumation, hep-th/9912093

*T. Kunihiro, for instance hep-th/9505166 and hep-th/9801196

*Polonyi, arxiv:hep-th/9409004, hep-th/9412042 and hep-th/9711061

*Tim R. Morris, hep-th/9802039

A: This is not answer to Your question in whole generality, I suggest You take a look in this simple but in my opinion meaningful example of renormalization in  simple situation: http://arxiv.org/abs/patt-sol/9709003 "Uses of Envelopes for Global and Asymptotic Analysis; geometrical meaning of renormalization group equation" Teiji Kunihiro and other papers by same author on  arxiv. 
While it may be generalized to more practical situations it gives also a hint what is it about in elementary way. 
A: Some very good references given so far.  I don't think I've seen this one mentioned yet - "Regularization, renormalization and dimensional analysis: Dimensional Regularization meets Freshman E&M"  by F Oleness and R Scalise.  Available here, it gives an extremely readable introduction to regularization and renormalization at the level appropriate for a very first exposure.
A: Hollowood's lecture notes "A Wilsonian Approach to Field Theory" are really nice.
If you're a mathematician interested in this stuff -- especially in renormalization as it appears in statistical mechanics -- you might want to try the Brydges' "Lectures on the Renormalization Group" in the book Statistical Mechanics in the "Ias/ Park City Mathematics" lecture series.  It discusses a few examples in some detail.  G. Battle's Wavelets and Renormalization is also a good place to look:  He discusses interacting scalar fields in a ton of detail.
There's also been some rigorous writing recently on renormalization in perturbative QFT, by Costello and by Borcherds, which might help people bridge the gap between the stat mech and more particle physics-y language.
A: The standard reference for many decades, which has not been improved upon in my opinion, is Kenneth Wilson's 1974 Reviews of Modern Physics article. It used to be required reading. It is a little old however.
A: My tutorial paper
 
Renormalization without infinities - a tutorial,
discusses renormalization and how to avoid the divergences on a much simpler level than quantum field theory.
See also Chapter B5: Divergences and renormalization of my theoretical physics FAQ.
A: I have written a pedagogical article about renormalization and renormalization group and I would be happy to have your opinion about it. It is published in American Journal of Physics. You'll find it also on ArXiv: A hint of renormalization.
B. Delamotte
A: My "eye-opener" about RG has been Amnon Aharony, see his book with Dietrich Stauffer "Introduction to Percolation theory".  The exposition of RG is totally focused on ideas,
and the (relatively) simple subject of the book -- a classical statistical lattice problem of percolation --  allows for very intuitive demonstrations, not requiring any QFT skills. Aharony became a postdoc of Michael Fisher a month after the Nobel-winning paper by Wilson & Fisher came out in 1972 (full text pdf), so the statistical physics roots of renormalization are very vivid in the book.
A: Renormalization is absolutely not just a technical trick, it's a key part of understanding effective field theory and why we can compute anything without knowing the final microscopic theory of all physics. One good online source that explains a nice physical example is Joe Polchinski's "Effective field theory and the Fermi surface" (and you can also look up the references therein). Also, pretty much any modern field theory textbook will explain the modern Wilsonian point of view on effective field theory. Some recent books that try to emphasize physical insight and not just calculation are those by Zee and by Banks.
A: And I have written another pedagogical article about renormalizations and IR divergences. I created a Google research group "QED Reformulation" and I run a blog on this subject. It is an alternative view of the problem, and I think, it is much more physical than the mainstream one. It is always useful to see the problem from different points of view ;-).
P.S. Se also this.
A: Regarding "providing unifying insight about renormalization in QFT, statistical physics or pure mathematics", this is what I tried to do in my detailed answer to Wilsonian definition of renormalizability
