My research focuses on modeling the dynamics and gravitational waves from coalescences of compact objects such as black holes and neutron stars driven by nonlinear gravity. An ever-increasing number of such signals has been measured in recent years by the LIGO and Virgo detectors, and already provided spectacular first discoveries and insights. The main scientific payoffs with gravitational waves are yet to come as detectors increase in sensitivity and explore larger swaths of frequencies, enabling deeper and higher precision studies. Black holes consist solely of warped spacetime according to General Relativity, while neutron stars comprise matter compressed by gravity to several times the density of an atomic nucleus, making them unique laboratories for subatomic physics in unexplored regimes. Gravitational waves provide a new probe of the structure of black holes and neutron star matter, and may also reveal the nature of dark matter or other elusive new fields from beyond standard model physics, and hints of quantum gravity. Extracting the rich information encoded in the gravitational wave signals requires cross-correlating the data with theoretical models that link gravitational-wave signatures to the fundamental source physics. Computing such models is challenging because one must determine the dynamical spacetime of the binary. My researchuses a variety of analytical approximation methods to develop accurate models and tests them against results from numerical relativity simulations. They serve as essential inputs for using gravitational waves to explore extremes of gravity and matter, potentially discover new objects and phenomena, and as a new cosmological messenger.