A topological dynamical system can support multiple invariant measures, and the space of such measures is a simplex. For dynamical systems with some hyperbolicity, this simplex is infinite-dimensional and has interesting properties. For example, its extreme points (the ergodic measures) are often dense in the simplex, which implies (among other things) that the set of extreme points is arc-connected. Moreover, the analytic and geometric properties of this simplex are related to the statistical properties of the underlying dynamical system. I will discuss this relationship in the classical case of uniformly hyperbolic systems and give some results concerning more general systems.