This is a revised and expanded version of the popular first edition, inspired by the work of Thom and Arnold on singularity theory. Such topics as umbilics, ridges and flexcords, all robust features of a smooth surface that are rarely treated in elementary courses on differential geometry, are considered here in detail. These features are of immediate relevance in modern areas of application, such as the interpretation of range data from curved surfaces and the processing of magnetic resonance and cat-scan images. The text is based on extensive teaching at Liverpool University, both to advanced undergraduate and beginning postgraduate students of mathematics. With these audiences in mind the author has included many examples and exercises to illustrate and illuminate important concepts. The wide applicability of the material covered here means that it will also appeal to scientists and engineers from a variety of other disciplines.