The purpose of this talk will be to describe some problems in topology whose solutions rely on geometric techniques. I will introduce the mapping class group of a space $X$, which is the group of homeomorphisms of $X$ up to isotopy. I will then outline Steve Kerckhoff's solution to the Nielsen realization problem, which shows that any finite subgroup of $Mod_g$, the mapping class group of a genus $g$ surface, can be realized as the automorphism group of a Riemann surface of genus $g$. The proof relies heavily on an understanding of the geometry of Teichmüller space $T_g$, and ties in to many natural geometric questions about this space which I hope to describe.