The Wess-Zumino-Witten (WZW) models are fundamental rational conformal field theories, and have a rich structure which has occasioned much interest. With regard to the further development of the formulation of this approach, as well as to the various applications of supersymmetric WZW models in superstring theories, the authors consider the question of whether one can generalise this covariant phase space formulation to the supersymmetric WZW models and discuss superparticles moving upon group manifolds. These systems share many of the important features of the supersymmetric WZW models. The WZW models are then discussed. It is shown that the full current algebras arise naturally for these models and the topological issues which arose in the bosonic case are found here with the same resolution. 22 refs.