Cosmological models with a contracting phase have generated interest recently, particularly in the framework of string theory. In this talk, I present some novel challenges and opportunities presented by these models. First, I discuss the possibility of violent, chaotic Mixmaster oscillations occuring in contracting universes, and suggest how this problem might be resolved. The stability of contracting universes may be used to constrain string models. Second, the transition from expansion to contraction is discussed. If such a transition occurs in our future, it can be constrained by studying dark energy and the variation of the fundamental constants. I discuss a novel feature of the perturbation theory of such models, and discuss applications to the cyclic and ekpyrotic cosmologies.