In this talk we will discuss the following result: if a bounded domain with C^{2} boundary covers a manifold which has finite volume with respect to either the Bergman volume, the K\"ahler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. The proof uses a variety of tools from Riemannian geometry and several complex variables including the squeezing function, Busemann functions, estimates on invariant distances, and a version of E. Cartan's fixed point theorem.