Finite N=2 supersymmetric σ-models

This report shows that N=4 supersymmetric models are on-shell ultraviolet finite to all orders in perturbation theory. This means that to all orders in perturbation theory, all ultraviolet divergences can be removed from all Greens functions by using only renormalization counterterms that arise from a (possibly non-linear) redefinition of the fields. The author demonstrates that the same result holds for N=2 supersymmetric models, provided that the target manifold M on which the model is defined is Ricci flat. The relation between these models and superstring theories formulated on non-trivial background spacetimes is detailed extensively elsewhere in these proceedings