Suggested reading for quantum field theory in curved spacetime

Quantum field theory (QFT) in curved spacetime is nowadays a mature set of theories quite technically advanced from the mathematical point of view.

There are several books and reviews one may profitably read depending on his/her own interests. I deal with this research area from a quite mathematical viewpoint, so my suggestions could reflect my attitude (or they are biased in favor of it).

First of all, Birrell and Davies' book is the first attempt to present a complete account of the subject. However the approach is quite old both for ideas and for the the presented mathematical technology, you could have a look at some chapters without sticking to it. Parker and Toms' recent textbook should be put in the same level as the classic by Birrel Davis' book in scope, but more up to date.

Another interesting book is Fulling's one ("Aspects of QFT in curved spacetime"). That book is more advanced and rigorous than BD's textbook from the theoretical viewpoint, but it deals with a considerably smaller variety of topics.

The Physics Report by Kay and Wald on QFT in the presence of bifurcate Killing horizons is a further relevant step towards the modern (especially mathematical) formulation as it profitably takes advantage of the algebraic formulation and presents the first rigorous definition of Hadamard quasifree state.

An account of the interplay of Euclidean and Lorentzian QFT in curved spacetime exploiting zeta-function and heat kernel technologies, with many applications can be found in a book I wrote with other authors ("Analytic Aspects of Quantum Fields" 2003)

A more advanced approach of Lorentzian QFT in curved spacetime can be found in Wald's book on black hole thermodynamics and QFT in curved spacetime. Therein, the microlocal analysis technology is (briefly) mentioned for the first time.

As the last reference I would like to suggest the PhD thesis of T. Hack http://arxiv.org/abs/arXiv:1008.1776 (I was one of the advisors together with K. Fredenhagen and R. Wald). Here, cosmological applications are discussed.

ADDENDUM. I forgot to mention the very nice lecture notes by my colleague Chris Fewster! http://www.science.unitn.it/~moretti/Fewsternotes.pdf

ADDENDUM2. There is now a quick introductory technical paper, by myself and I.Khavkine, on the algebraic formulation of QFT on curved spacetime: http://arxiv.org/abs/1412.5945 which in fact will be a chapter of a book by Springer.

To complement Valter Moretti's excellent answer, I would like to offer a much less rigorous and much more intuitive, physically-oriented alternative: Mukhanov (& Winitzki) - Introduction to Quantum Effects in Gravity.

This is a very interesting book, not just for the QFT in curved spacetime, but because it aims to give something like a unified view of physics, starting from classical mechanics and proceeding all the way up to QFT in curved spacetime (Unruh effect, Hawking radiation, heat kernel etc. are treated---though not in great depth) in a more-or-less coherent way. I found it to be very enlightening, and it is also a tad easier than the referenced mentioned by Valter. It is also interesting to note that a draft (very similar to the final version; I compared them myself) is available for free online.