The Zimmer program seeks to classify smooth actions of arithmetic groups, like SL(n,Z), on compact manifolds. Separately, the Nielsen realization problem asks when a subgroup of a mapping class group Mod(M) can be realized by a group of diffeomorphisms of M. In many natural situations, the mapping class group is closely related to an arithmetic group, and the realization problem is tied to the Zimmer program. I will discuss examples of this connection and describe some recent results and open questions.