Adiabatic perturbation theory is a general scheme that allows for the mathematically rigorous derivation of effective equations in quantum mechanical slow-fast systems. In this lecture I explain how to justify and compute corrections to the time-dependent Born-Oppenheimer approximation. In the second part I present some recent results (jointly with Volker Betz) on computing the dynamics of exponentially small non-adiabatic transitions between different electronic energy surfaces.