We review the concept and applications of a semiclassical (ε-classical or pseudoclassical) approximation to the resonant dynamics of an atom "kicked" by a pulsed, periodic potential. This powerful method allows us to derive analytical results in the deep quantum limit of the kicked rotor. Additionally, classical phase space portraits may be used to represent the dynamics even though the system is fundamentally quantum mechanical. The technique has been successfully adapted for systems including noise and decoherence, as well as systems for which the initial state is a trivial quantum superposition (leading to directed transport at quantum resonance). For almost a decade, theoretical investigations and experimental investigations have been proceeding hand-in-hand in this field, which has been stimulated regularly by experimental progress in controlling driven dynamical systems. Here, we review both theoretical and experimental advances, which in turn may inspire future applications of the presented pseudoclassical method.