Thurston revolutionized the study of diffeomorphisms of Riemann surfaces with a myriad of tools: hyperbolic geometry, geometric group theory, Teichmuller theory, and much more. One goal of this work is to see how much of his vision persists in higher dimensions. In this direction, we will discuss a smooth analog of the Birman-Hilden theory for certain 4-manifolds, and its applications to Nielsen realization problems. Time permitting, we will also discuss related joint work with Seraphina Lee and Tudur Lewis on the geometry of other 4-manifold mapping class groups, via moduli and reflection group theory.