Part I attempts to review the background, basic sources of data, concepts, and methodology to be employed in the study of perceptrons. In Chapter 2, a brief review of the main alternative approaches to the development of brain models is presented. Chapter 3 considers the physiological and psychological criteria for a suitable model, and attempts to evaluate the empirical evidence which is available on several important issues. Chapter 4 contains basic definitions and some of the notation to be used in later sections are presented. Parts II and III are devoted to a summary of the established theoretical results obtained to date. Part II (Chapters 5 through 14) deals with the theory of three-layer series-coupled perceptrons, on which most work has been done to date. Part III (Chapters 15 through 20) deals with the theory of multi-layer and cross-coupled perceptrons. Part IV is concerned with more speculative models and problems for future analysis. Of necessity, the final chapters become increasingly heuristic in character, as the theory of perceptrons is not yet complete, and new possibilities are continually coming to light.