  Extreme mass ratio inspirals (EMRIs) show a strong separation of timescales,
with the time characterizing inspiral, $T_{\rm i}$, much longer than any time
$T_{\rm o}$ characterizing orbital motions. The ratio of these timescales
(which is essentially an EMRI's mass ratio) can be regarded as a parameter that
controls a perturbative expansion. Here we describe the value and limitations
of an "adiabatic" description of these binaries, which uses only the leading
terms arising from such a two-timescale expansion. An adiabatic approach breaks
down when orbits evolve through resonances, with important dynamical and
observational consequences. We describe the shortfalls of an approach that only
includes the adiabatic contributions to EMRI evolution, and outline what must
be done to evolve these systems through resonance and to improve our ability to
model EMRI systems more generally.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena