At the simplest level, gravitational sources are considered to be point-like and in solitude, with a radial force that falls off as r^-2. In reality, all astrophysical objects aside from black holes are extended in space, and can be deformed by the tidal forces arising from the proximity of companion objects with large average densities. When these forces are weak, the response of an object to a tide can be through a decomposition into basis functions, but this approach fails when the tide is strong enough to deform an object by a distance equal to its own size. Under these circumstances, a hydrodynamical representation of the object is required to understand the true tidal response.

In this thesis, we present a number of examples of physical systems in which tides dominate the dynamics. First, we consider the case of a star that encounters a supermassive black hole (SMBH) in a deeply penetrating encounter, resulting in a dramatic compression that produces shocks that would be observable in the X-ray. Second, we present the results of hydrodynamical simulations that demonstrate a new mechanism for igniting Type Ia supernovae from binary systems composed of two white dwarfs undergoing Roche-lobe overflow. Third, we investigate the survival prospects of giant planets that have been scattered into highly eccentricity orbits and are exposed to a strong tide applied by their parent star. Fourth, we systematically map the fallback rate resulting from the tidal disruptions of stars by SMBHs. Finally, we use what we have learned about the feeding rate to model determine the highest-likelihood model for an observed prototypical tidal disruption event.