  We explain why the conventional argument for deriving the time-dependent
Born-Oppenheimer approximation is incomplete and review recent mathematical
results, which clarify the situation and at the same time provide a systematic
scheme for higher order corrections. We also present a new elementary
derivation of the correct second-order time-dependent Born-Oppenheimer
approximation and discuss as applications the dynamics near a conical
intersection of potential surfaces and reactive scattering.
