I will describe some recent work on the first order theory of homeomorphism groups of manifolds. I will discuss a new result which shows that the homeomorphism groups of two compact manifolds are elementarily equivalent if and only if the two manifolds are homeomorphic, which resolves an old conjecture of Rubin. I will then describe some of the expressive power of the language of groups in the theory of homeomorphism groups, with implications for the subgroup structure of homeomorphism groups, and for the descriptive set theory of these groups. This represents joint work with S. Kim and J. de la Nuez Gonzalez.