Geometrical structure and ultraviolet finiteness in the supersymmetric sigma-model

A complete geometrical classification of supersymmetric sigma-models is given. Extended supersymmetry requires covariantly constant complex structures, and Kahler and hyperkahler manifolds play a special role. As an application of the classification, it is shown that a particular class of these models is on-shell ultraviolet finite to all orders in perturbation theory. (orig.)