This book is based on a course of sixteen lectures recently given at the University of Oxford to an audience of undergraduate and graduate students. About half the material is classical but the remainder is not so well-known. Indeed much of it is published nere for the first time.

The theory of uniform spaces was developed by Weil [25] and others in the thirties. The classic account of the subject is in Chapter II of Bourbaki [2]. However, the majority of the more recent text books on topology contain at least an outline of the theory. Moreover there are several specialized monographs,; such as Isbell [7], Page [19], Roelcke and Dierolf [21], amongst others.

For students of mathematics the transition from metric spaces to general topological spaces is a major step which many find extremely difficult. Uniform spaces make an excellent intermediate stage. I have therefore written the first two chapters in such a way that they can be read by a student with no knowledge of topology. The second two chapters assume a basic knowledge of topology and are aimed at showing how the uniform world and the topological world are related.

The monographs on the theory mentioned above are written mainly with the needs of analysts in mind. Rather than go over the same ground again I have chosen to explore a different aspect of the theory.