  Much of the success of gravitational-wave astronomy rests on perturbation
theory. Historically, perturbative analysis of gravitational-wave sources has
largely focused on post-Newtonian theory. However, strong-field perturbation
theory is essential in many cases such as the quasinormal ringdown following
the merger of a binary system, tidally perturbed compact objects, and
extreme-mass-ratio inspirals. In this review, motivated primarily by
small-mass-ratio binaries but not limited to them, we provide an overview of
essential methods in (i) black hole perturbation theory, (ii) orbital mechanics
in Kerr spacetime, and (iii) gravitational self-force theory. Our treatment of
black hole perturbation theory covers most common methods, including the
Teukolsky and Regge-Wheeler-Zerilli equations, methods of metric
reconstruction, and Lorenz-gauge formulations, casting them in a unified
notation. Our treatment of orbital mechanics covers quasi-Keplerian and
action-angle descriptions of bound geodesics and accelerated orbits, osculating
geodesics, near-identity averaging transformations, multiscale expansions, and
orbital resonances. Our summary of self-force theory's foundations is brief,
covering the main ideas and results of matched asymptotic expansions, local
expansion methods, puncture schemes, and point particle descriptions. We
conclude by combining the above methods in a multiscale expansion of the
perturbative Einstein equations, leading to adiabatic and post-adiabatic
evolution and waveform-generation schemes. Our presentation includes some new
results but is intended primarily as a reference for practitioners.
