Dynamical systems can roughly be divided into those which we do understand, and  
those which we don't. It has been assumed since the 1970's that hyperbolic  
systems were included among those which we do. However, in the last ten years,  
natural questions posed about the geometry of hyperbolic systems made it clear  
that often, they were not all that well understood! This talk we will discuss  
some of these natural questions and the new geometric ideas introduced to solve  
them. These ideas have had major applications in the last five years towards  
proving a geometric classification (or rigidity) for certain classes of  
hyperbolic systems.