The main difficulty of perturbative quantum gravity (QG) in D=4 dimensions is the conflict between renormalizability and unitarity of the theory. The simplest version of QG is based on General Relativity and turns out to be nonrenormalizable. One can construct renormalizable and even superrenormalizable versions of QG by introducing higher derivatives. In the local versions of higher derivative models of QG there are always unphysical higherderivative massive unphysical ghosts. One can construct non-polynomial in derivatives (or, equivalently, nonlocal) models of QG, which have no ghosts at the tree level. However, taking loop corrections into account one meets infinite amount of ghost-like complex states. The theories of both local and nonlocal types attracted a lot of attention in the last years and our purpose it to present a brief review of the problems and perspectives of these models, according to the present-day understanding.