During the last few years, considerable interest has been focused on the phase that waves accumulate when the equations governing the waves vary slowly. The recent flurry of activity was set off by a paper by Michael Berry, where it was found that the adiabatic evolution of energy eigenfunctions in quantum mechanics contains a phase of geometric origin (now known as Berry's phase') in addition to the usual dynamical phase derived from Schr dinger's equation. This observation, though basically elementary, seems to be quite profound. Phases with similar mathematical origins have been identified...