  Much of the progress in the gravitational self-force problem has involved the
use of singular perturbation techniques. Yet the formalism underlying these
techniques is not widely known. I remedy this situation by explicating the
foundations and geometrical structure of singular perturbation theory in
general relativity. Within that context, I sketch precise formulations of the
methods used in the self-force problem: dual expansions (including matched
asymptotic expansions), for which I identify precise matching conditions, one
of which is a weak condition arising only when multiple coordinate systems are
used; multiscale expansions, for which I provide a covariant formulation; and a
self-consistent expansion with a fixed worldline, for which I provide a precise
statement of the exact problem and its approximation. I then present a detailed
analysis of matched asymptotic expansions as they have been utilized in
calculating the self-force. Typically, the method has relied on a weak matching
condition, which I show cannot determine a unique equation of motion. I
formulate a refined condition that is sufficient to determine such an equation.
However, I conclude that the method yields significantly weaker results than do
alternative methods.
