This textbook covers all the theoretical aspects of real variable analysis which undergraduates reading mathematics are likely to require during the first two or three years of their course. It is based on lecture courses which the author has given in the universities of Wales, Cambridge and London. The subject is presented rigorously and without padding. Definitions are stated explicitly and the whole development of the subject is logical and self-contained. Complex numbers are used but the complex variable calculus is not. 'Applied analysis', such as differential equations and Fourier series, is not dealt with. A large number of examples is included, with hints for the solution of many of them. These will be of particular value to students working on their own.