  Over the next decade, third-generation interferometers and the space-based
LISA mission will observe binaries in galactic centers involving supermassive
black holes with millions of solar masses. More precise measurements of more
extreme events that probe stronger gravitational fields can have a tremendous
impact on fundamental physics, astrophysics, and cosmology. However, at the
galactic scale, accretion disks, dark matter halos, and dense populations of
compact objects can interact gravitationally with coalescing bodies. The role
these astrophysical structures play in the evolution and gravitational-wave
signature of binary systems remains largely unexplored and previous studies
have often relied on ad-hoc Newtonian approximations. In this thesis, we aim to
improve this picture. We study how tidal deformations of matter surrounding
black holes can mask off deviations from General Relativity. We also explore
the deep connection between light rings -- closed orbits of massless particles
-- and the proper oscillation modes of compact objects. We show that
independently of the presence of an environment, the light ring controls the
late-time appearance of infalling matter to distant observers and how the final
black hole formed in a collision relaxes to stationarity. Finally, we develop
the first fully-relativistic framework capable of studying gravitational wave
emission in non-vacuum environments. We apply it to galactic black-hole
binaries surrounded by a dark matter halo and observe the conversion between
matter and gravitational waves. This coupling results in significant changes in
the energy flux emitted, which could help constrain the properties of galactic
matter distributions.
