Let G be a compact Lie group, and let S be a surface group. The character variety is the set of homomorphisms from S to G, modulo conjugation. Since the mapping class group is the outer automorphism group of S, it acts on the character variety by precomposition. It is natural to ask about stationary measures and orbit closures for this action.

The breakthrough work of Benoist and Quint and subsequent developments gives a new set of tools for attacking this problem. I will give a progress report and describe the difficulties in the general case.