Einstein's theory of General Relativity predicts a Cosmos with black holes and gravitational waves. Although neither black holes nor gravitational waves have been directly detected, their presence is already felt throughout the Universe. This decade will witness observations for which gravitational waves are the messengers that deliver information in exquisite detail about astrophysical phenomena, among them the collision of two black holes, a system completely invisible to the eyes of traditional telescopes. Models that predict gravitational wave signals from likely sources are crucial for the success of this endeavor. Modeling sources of gravitational radiation requires solving the Eintein equations of General Relativity using powerful computer hardware and sophisticated numerical algorithms. In this talk I will review these challenges, how we have overcome them, and what we have learned along the way. Our predictions of the gravitational waves from the black holes collisions is one pivotal step in ushering in the new era of gravitational-wave astronomy.