Abstract: Perelman's celebrated stability theorem showed that if a convergent sequence of Alexandrov spaces does not drop in dimension on passing to the limit, then the objects in the tail of the sequence are homeomorphic to the limit. In this talk, the theorem will be extended to an equivariant setting. As an application, it will be shown that two classes of Riemannian orbifolds, defined by geometric and spectral constraints, are finite up to orbifold homeomorphism.