  We introduce an adiabatic perturbation theory for quantum systems with
degenerate energy spectra. This perturbative series enables one to rigorously
establish conditions for the validity of the adiabatic theorem of quantum
mechanics for degenerate systems. The same formalism can be used to find
non-adiabatic corrections to the non-Abelian Wilczek-Zee geometric phase. These
corrections are relevant to assess the validity of the practical implementation
of the concept of fractional exchange statistics. We illustrate the formalism
by exactly solving a time-dependent problem and comparing its solution to the
perturbative one.
