Abstract: The main difficulty of perturbative quantum gravity (QG) in 4d is the conflict between renormalizability and unitarity. The simplest version of QG is based on General Relativity and is non-renormalizable. One can construct renormalizable and even superrenormalizable versions of QG by introducing higher derivatives, but then one has to deal with the unphysical higher derivative massive ghosts. At the same time polynomial superrenormalizable versions of QG have certain attractive features, such as unambiguous and exactly calculable beta-functions and possible Lee-Wick type unitarity in case when all extra poles are complex. The non-polynomial models of QG have no ghosts at the tree level, but taking loop corrections into account one meets infinite amount of ghost-like complex states. In the IR one can prove that the modified Newton limit in all these models is singularity-free. After a survey of all these issues some list of unsolved interesting problems will be discussed.