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Extreme mass-ratio inspirals, in which solar-mass compact bodies spiral intosupermassive black holes, are an important potential source for gravitationalwave detectors. Because of the extreme mass-ratio, one can model these systemsusing perturbation theory. However, in order to relate the motion of the smallbody to the emitted waveform, one requires a model that is accurate onextremely long timescales. Additionally, in order to avoid intractabledivergences, one requires a model that treats the small body as asymptoticallysmall rather than exactly pointlike. Both of these difficulties can be resolvedby using techniques of singular perturbation theory. I begin this dissertationwith an analysis of singular perturbation theory on manifolds, including thecommon techniques of matched asymptotic expansions and two-timescaleexpansions. I then formulate a systematic asymptotic expansion in which themetric perturbation due to the body is expanded while a representativeworldline is held fixed, and I contrast it with a regular expansion in whichboth the metric and the worldline must be expanded. This results in anapproximation that is potentially uniformly accurate on long timescales. Theequation of motion for the body's fixed worldline is determined by performing alocal-in-space expansion in the neighbourhood of the body. Using this localexpansion as boundary data, I construct a global solution to the perturbativeEinstein equation. To concretely characterize orbits, I next devise arelativistic generalization of the Newtonian method of osculating orbits.Making use of this method and two-timescale expansions, I examine the utilityof adiabatic approximations that can forgo an explicit calculation of theforce.

Motion of small bodies in general relativity: foundations and implementations of the self-force