To find gravitational waves in LIGO's noisy data and to extract the information the waves carry, we need theoretical insights into their waveforms. In some cases, these insights can come from analytic solutions of Einstein's general relativity equations, but in many cases we must rely on numerical simulations of the sources. For some sources (e.g., the accretion-induced collapse of a white dwarf star), the simulations can be based on Newtonian fluid mechanics; but for most sources (e.g., the merger of two black holes or the tidal destruction of a neutron star by a black hole) the simulations require numerical solution of Einstein's equations --- "numerical relativity". The progress, to date, in such simulations, both Newtonian and relativistic, has been meager compared to LIGO's long-term needs. Thorne will describe the simulations that are needed, the obstacles to carrying them out, and their current status.