I will talk about some results on regularity of stationary measures of random dynamical systems. Under the assumption of absence of almost sure invariant measures we prove Holder continuity or log-Holder continuity of all stationary measures depending on estimates for the regularity of random maps. The methods I am going to present can also be used in the non-stationary case. This talk is based on a recent joint work with Anton Gorodetski and Victor Kleptsyn.