Abstract: Mathematical cosmology is the study of solutions to Einstein's equations with cosmological boundary conditions. (These spacetimes are in contrast to those containing black holes with the asymptotically flat boundary conditions appropriate to isolated systems.) While most cosmological spacetimes do not describe the actual universe, they do provide arenas to explore properties of strong field gravity and to test algorithms for computer simulations of Einstein's equations. One area of study concerns singularities and cosmic censorship. Powerful theorems state that evolution of regular, generic initial data by Einstein's equations will lead to some sort of pathology (singularity) if the gravitational field becomes strong enough. The cosmic censorship conjectures claim that such singularities in generic physical systems will be hidden from us by black hole event horizons and / or could not be detectable by us until we are unfortunate enough to fall into them. Cosmological spacetimes provide an excellent arena for the study of singularities and cosmic censorship. Numerical simulations have proven to be a valuable tool to explore these spacetimes especially in collaboration with mathematical analysis. While computers cannot handle infinite or undefined values, their ability to evolve complicated nonlinear equations allows them to yield insight into the approach to pathological behavior in Einstein's equations. In contrast, mathematical analyses can prove theorems that cover large classes of spacetimes. I shall discuss examples of spacetimes where numerical simulations have guided the development of theorems and also examples where simulations can provide guidance for future theorems.