This is the proceedings of the 1983 AMS Short Course given at Denver, Colorado. Computer communications is characterized by rapid technological advances presenting problems of a theoretical nature that are often very difficult to solve. They range from those that arise on a single chip, where communication among thousands of elements on a chip is influenced by electrical properties, to those that arise when human beings communicate with databases where the logical aspects of communications play a more important role. A variety of mathematical methods is needed to attempt to solve such problems; they vary from partial differential equations to temporal or modal logic. The article on ""Diffusion Approximation"" is probably one of the best reviews of this field. The book is unique in having contributors from a variety of different fields of computer communications. All of the articles are of high research value and are self-contained.