  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, presenting them in a new
consistent and self-contained form. 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 schemes. Our presentation is intended
primarily as a reference for practitioners but includes a variety of new
results. In particular, we present the first complete post-adiabatic
waveform-generation framework for generic (nonresonant) orbits in Kerr.
