One of the most remarkable discoveries in elementary particle physics has been that of the existence of the complex plane. From, the early days of dispersion relations onwards the analytic approach to the subject has proved a most useful tool. In this monograph we give an account of two aspects of this approach. The first topic, dealt with in chapters 2 and 3, is the study of perturbation theory as a model with, heuristic value as a guide to what we may expect to be true in the actual physical theory. The (second topic, dealt with in chapter 4, is the exploration of what we may hope will prove to be the actual physical theory, that is to say the investigation of the properties of the analytic S-matrix.