Large-scale computer simulations are increasingly crucial in explaining astrophysical phenomena. Just one recent example is the stunning agreement of the full 3-d solution of Einstein's equations for colliding black holes with the observed signal from LIGO. For the past 50 years, the dominant computer method for solving these kinds of equations has remained essentially unchanged. To keep up with continuing advances in observation, simulations will require more fidelity and higher accuracy.  One might think that with exascale machines becoming available in the next 5 years, this will be easy. This is not true: I will explain why Moore's Law is broken, and how the next generation of supercomputers will instead get their power by having millions of processors. Current codes will not be able to use these machines efficiently. I will describe new methods for harnessing the power of such exascale computers to solve some of the largest problems in astrophysics and other areas of science.