A real projective structure on a compact manifold is a maximal atlas of charts whose images are in real projective space and whose transition functions are restrictions of projective transformations. In this talk I'll give an introduction to the Hilbert metric on a manifold with a real projective structure and give a number of applications. My plan is to first describe a theorem of Kobayashi which gives a sufficient condition for the developing map to be a diffeomorphism onto a convex domain. Then I plan on describing work of Benoist which gives further restrictions on the developing map when the fundamental group is Gromov hyperbolic. I will end the talk with some open questions. Speaker(s): Andrew Zimmer (University of Chicago)