Difficulties in quantizing gravity

(i) (mathematical) No consistent interaction relativistic quantum field theory is known in 4 dimensions.

(ii) (theoretical) The accepted ways to avoid divergences in expressions for scattering amplitudes that work in simpler theories all fail because of the lack of renormalizability. See, e.g., the references in Section 2.2 of http://relativity.livingreviews.org/Articles/lrr-2002-5/

(iii) (theoretical) The theories for which a (perturbatively) finite scattering theory is available have not been related quantitatively to the established theories. A convincing classical limit (to general relativity), nonrelativistic limit (to a multiparticle Schroedinger equation with Newtonian interaction), and low energy limit (at currently accessible energies no new particles apart from the graviton) would be needed.

(iv) (conceptual) The three limits pose severe constraints on possible quantum gravity theories, and it requires much imagination to come up with a conceptual basis in which these limit make sense and are tractable.

(v) (experimental) Quantum effects in gravity are so weak that no experiments sensitive to quantum effects are in reach in the near future, and the data from astronomy that may cast light on quantum gravity are scarce. (Quantum gravity is not demanded by unexplained data but only by the quest for consistency with particle physics.)

But see also
Is quantum mechanics compatible with general relativity?

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ