General Relativity research and QFT in curved spacetime A naive question: 
Are these subjects, i.e. classical GR and QFT in curved spacetime, being worked upon much anymore? 
Who is researching this and what are the problems within these fields? Any sources are greatly appreciated.
 A: For the state of the art of research in general relativity, have a look at this site:


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*living reviews in relativity (list by subject).


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:


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*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:


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*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:


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*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
A: A few topics comes to mind on QFT on curved spacetime are:


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*Quantum inequality, which addresses bounds on quantum violation of energy conditions. Just type in Larry Ford, Tom Roman or Chris Fewster for recent development.

*DeSitter instabilities, which addresses whether DeSitter invariant vacua exist for interacting field theory. Perhaps a good starting point is a following paper by Polyakov.
http://arxiv.org/abs/0912.5503
and a paper by Higuchi, Marolf and Morrison. 
http://arxiv.org/abs/1107.2712


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*A rigorous existence of interacting field theory. I think a paper by Hollands is a good reference.


http://lanl.arxiv.org/abs/0705.3340
A: There are several interesting approaches to dark matter in classical gravity. For explaining rotation curves in spiral galaxies without appealing to dark matter check out Cooperstock 
http://lanl.arxiv.org/abs/astro-ph/0507619
and subsequent papers (which I am less familiar with).
Also, there is an interesting idea generically called "exotic smoothness", which (among other things) might reproduce lensing experiments that imply the existence of dark matter. A good introduction by Carl Brans can be found here: 
http://lanl.arxiv.org/abs/gr-qc/9212003
He also coauthored a textbook on the subject with Torsten Asselmeyer-Maluga, Exotic Smoothness and Physics: Differential Topology and Spacetime Models published by World Scientific. 
A: There are tons of research going on in the classical general relativity. One particularly active subarea is the so-called "numerical relativity", in which the evolution of the spacetime is simulated in computers. 
Another smaller subarea, which I mention here because I'm personally interested in it, is the "fluid-gravity correspondence" which is the study of the hydrodynamic behavior of the spacetime close to the black holes. For example, the Navier-Stokes equation arises naturally as a certain limit of the Einstein equation.
A: From a very biased point of view of a mathematician, there are lots of open problems in classical general relativity. I enumerated some at this post at Math.Stackexchange. Note that some of those are considered "solved" by (members of) the physics community (the "No-hair theorem", for example, on the uniqueness of Kerr-Newman black holes); but mathematicians sometimes like to poke holes in arguments. 
A: One field which uses part of the formalism is relativistic quantum information.  You might want to check out 
http://www.isrqi.org/
Some of the problems being researched are:
1. Quantum information on curved backgrounds
arXiv 1108.3896


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*Vacuum entanglement
arXiv 1105.1192

*Closed time like curves 
arXiv 1003.1987

*Entanglement and quantum information for accelerating observes 
quant-ph/0410172
There is  a 2003 review on the subject but it does not include too much about QFT on a curved ST.
http://arxiv.org/abs/quant-ph/0212023
