For the state of the art of research in general relativity, have a look at this site:
For research of QFT in curved spacetime there are actually two quite different approaches, Lagrangian QFT and axiomatic QFT. I don't know much about the former, I only skimmed this recently published texbook:
- Leonard Parker, David Toms: "Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity", (Cambridge Monographs on Mathematical Physics)
So at least two people are still active in that area :-)
People in axiomatic QFT have been busy to generalize the Haag-Kastler axioms to curved spacetimes. There are several concepts that obviously have to be generalized, like the spectral condition that uses the Poincare group. This has been accomplished several years ago and now people are generalizing theorems and concepts.
A good place to start is this textbook:
- Wald, Robert M.: "Quantum field theory in curved spacetime and black hole thermodynamics". Univ. of Chicago Press 1994.
which includes a derivation of Hawking radiation and the Unruh effect.
Recently published review papers are:
This post has been migrated from (A51.SE)
Romeo Brunetti, Klaus Fredenhagen: "Quantum Field Theory on Curved Backgrounds", arXiv
Robert M. Wald: "The Formulation of Quantum Field Theory in Curved Spacetime", arXiv
Robert M. Wald: "The History and Present Status of Quantum Field Theory in Curved Spacetime", arXiv