Abstract
We investigate non-perturbative contributions to ultraviolet divergences in several field theories. Although we concentrate on finite theories with extended supersymmetry, we also present some results for asymptotically free theories. The
N = 2 supersymmetric σ models that we study at the end of the paper are important in the theory of superstrings. Unfortunately, while we are able to rule out instanton contributions to the β functions of these theories, we do not have a complete non-perturbative proof that they are conformally invariant. Sigma models with
N = 4 supersymmetry are shown to be both perturbatively and non-perturbatively finite.