I have learned (the basics) of how one can incorporate the principles of special relativity to quantum mechanics to obtain quantum field theory. Can the same be done with GR and QM to obtain a new theory? If not, what happends if one tries, what exactly goes wrong? Is there a mathematical inconsistency in any theory that combines them, such that either current GR or QM must be modified, or is it just that we dont know how to yet?
General Relativity + Quantum Mechanics = String theory.
The things that go wrong in a non string approach;
All these problems might be surmounted, but this requires a new idea, like loops. In 2d and 3d gravity, the same problems occur, but they are resolved completely by the topological nature of gravity. Loops give an interesting idea for 4d, but it is incomplete.
The way to solve all four problem at once is to move to an S-matrix theory for gravity. This idea, originally proposed by Heisenberg, but best attributed to Geoffrey Chew and Stanley Mandelstam, means that you define the theory asymptotically in terms of scattering states, not in terms of microscopic fields. This works to produce string theory.
Well, some people might say that string theory is the answer. If one takes the usual framework of quantum field theory (say, the standard model), we run into trouble if we want to add gravity. I think one can prove that it doesn't work (it's not renormalizable), and it's not just that we don't know how to do it. But finding the right theory that includes quantum mechancis and gravity is one of the big open problems in physics.
Someone more knowledgeable on the subject can surely point out exactly what the difficulties are.