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 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 either for ideas and for the mathematical technology, you could have a look at some chapters without sticking to it.
The much more recent book by Mukhanov and Winitizki can be considered a physically modern version of it (at least referring to some topics).
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
This post imported from StackExchange Physics at 2014-05-01 13:25 (UCT), posted by SE-user V. Moretti