Supercomputer simulations allow us to effectively compress the vast cosmological timescales that apply to the largest structures in our Universe down to weeks of virtual time. This gives us a completely new view of the Universe from what is available to us by observation alone. Furthermore, it is the closest thing to a physics lab that is available to the astrophysicist. However, a useful virtual laboratory of this type requires the ability to process an incredible number of floating point calculations. I will discuss the use of supercomputing techniques in celestial mechanics and cosmogony simulations, focusing on the various approaches that allow parallel computers to be harnessed in this field. Both small and large numbers of bodies can be simulated on these parallel machines, where the algorithms used at either extreme differ substantially. Examples and a few selected results of these studies will be presented throughout.