The groups Homeo(M) and Diff(M) of homeomorphisms or diffeomorphisms of a manifold M have many striking parallels with finite dimensional Lie groups. In this talk, I’ll describe some of these, and explain joint work with Lei Chen, that gives a structure theorem for actions of homeomorphism and diffeomorphism groups on other spaces, analogous to some classical results for actions of locally compact Lie groups. As applications, we answer many concrete questions towards classifying all actions of Diff(M) on other manifolds (many of which are nontrivial, for instance Diff(M) acts naturally on the unit tangent bundle of M…) and resolve several threads in a research program initiated by Ghys. I’ll aim to give both a broad overview and several basic applications in the talk.