General relativity is the fundamental theory of gravity, which is governed by an extremely complex set of coupled, nonlinear, hyperbolic-elliptic partial differential equations. General solutions to these equations, needed to fully understand their implications as a fundamental theory of physics, are elusive. Additionally, the astrophysics of compact objects, which requires Einstein’s theory of general relativity for understanding phenomena such as black holes and neutron stars, is attracting increasing attention. The largest parallel supercomputers are finally approaching the speed and memory required to solve the complete set of Einstein’s equations for the first time since they were written over 80 years ago, allowing one to attempt full 3D simulations of such exciting events as colliding black holes and neutron stars. In this paper we review the computational effort in this direction, and discuss a new 3D multipurpose parallel code called ‘Cactus’ for general relativistic astrophysics. Directions for further work are indicated where appropriate.