The regret theory of choice under uncertainty is known to admit intransitivities in preference relations. In this paper, the stochastic dominance properties of the theory are examined. It is shown that the usual definition of first stochastic dominance is not satisfied by regret-theoretic preferences and that, in general, violations of first stochastic dominance are not merely permitted but required. An exact characterization of the stochastic dominance rule corresponding to regret-theoretic preferences is presented. This concept is weaker than the usual definition, but stronger than the notion of statewise dominance in which one prospect yields a preferred outcome with probability 1. Theories of choice under uncertainty have proliferated in recent years, after a long period in which the expected utility (EU) approach of Von Neumann and Morgenstern (1944) was pre-eminent. The breakdown of the dominance in EU theory reflects the increasing body of evidence that the choices made by many people systematically violate the predictions of the theory. The axiomatic foundations of the theory, and in particular the "independence axiom" have also received severe questioning, ever since the famous critique of Allais (1953).