Summary: In many applications, systems of reaction-diffusion equations arise in which the nature of the nonlinearity in the reaction terms renders ineffective the standard techniques (such as invariant sets and differential inequalities) for establishing global existence, boundedness, and asymptotic behavior of solutions. In this paper we prove global existence and uniform boundedness for a class of reaction- diffusion systems involving two unknowns in which an a priori bound is available for one component as long as solutions exist. Among this class of systems is the so-called Brusselator, a model from the study of instabilities in chemical processes.