Gravitational waves---ripples of spacetime curvature---are poised to open a new window on the universe: the Advanced Laser Interferometer Gravitational-wave Observatory (Advanced LIGO), scheduled for completion in 2015, is expected to detect between 1 and 1000 gravitational waveforms per year. Among the most important sources for Advanced LIGO are collisions of two black holes (which can radiate as gravitational waves more energy than a supernova) and of a black hole and a neutron star (which can radiate both gravitational and electromagnetic waves, raising the prospect of ``multi-messenger astronomy.''). Finding these gravitational waves in noisy LIGO data requires accurate predictions of the expected waveforms, but because all analytic approximations break down, the waves can only be predicted using numerical simulations. In this talk, I will discuss recent progress and challenges in supercomputer simulations of black hole-black hole and black hole-neutron star collisions, focusing on the particularly challenging case of black holes that are very rapidly rotating.