Let M be a manifold, and let Mod(M) be its mapping class group. The Nielsen realization problem for diffeomorphisms asks, “Can a given subgroup G<Mod(M) be lifted to the diffeomorphism group Diff(M)?” This question about group actions is related to a question about flat connections on fiber bundles with fiber M. In the case M is a closed surface, the answer is “yes" for finite G (by work of Kerckhoff) and “no" for G=Mod(M) (by work of Morita). For most infinite G<Mod(M), we have no idea. I will discuss some obstructions that can be used to show that certain groups don’t lift. Some of this work is joint with Nick Salter.