Miraculous ultraviolet cancellations in supersymmetry made manifest

We develop the formalism for quantization with manifest linearly realized extended supersymmetry. Together with the use of the background field method, this gives rise to a general non-renormalization theorem that explains all of the 'miraculous' cancellations observed to date in supersymmetric field theories, and shows in which cases they will be continued to all orders of perturbation theory. In particular, we present the details of N = 2 superfield perturbation theory and show that in renormalizable models in four dimensions, the only infinity that can arise is at the one-loop order in the Yang-Mills coupling constant. The finiteness of the N = 4 supersymmetric Yang-Mills theory is a particular consequence of this result. (orig.)