Due to their binary nature, the Walsh functions have considerable advantages over the traditional sinusoidal functions used in Fourier analysis when the computations are carried out by a general purpose binary digital computer. The important properties of the Walsh functions which illustrate these advantages are examined and developed. The Walsh transform and spectrum are presented in relation to the problem of function approximation, and various computational procedures for effecting the transform are explained. The unconventional 'logical' transform is developed from the Walsh transform, and there is a discussion on the subject of interpreting the resulting spectrum. There are other functions, such as the Haar functions, which are closely related to the Walsh functions, and their advantages are indicated. The process of shape analysis is dealt with in terms of its relation to the more widely treated problem of pattern recognition. An application of shape analysis, using Walsh functions, to a study of leaf shapes is illustrated by experimental results. A completely different approach to shape analysis is taken in the chapter on Pattern Generation and Simulation of Growth Processes. Other applications of Walsh functions, particularly of the 'logical' transform, are discussed in the final chapter. Throughout, tested computer programs are used to provide examples, back up conjectures, and generally illustrate numerous points in the text.