Randomly interacting p-state Potts spins may freeze into a Potts-glass phase in which the Potts symmetry is unbroken, on the average. The mean-field theory of this phase transition is presented. Unlike the spin-glass case, there exist two distinct Potts-glass phases that differ in the nature of the correlations among the many degenerate ground states of the system. For pg4, the transition from the disordered phase is unusual: The freezing occurs discontinuously but without latent heat. Similar results hold for mean-field quadrupolar-glass models.