This book charts the progress in the theory of magnetic impurities since the late 50s, from the early developments leading to the Kondo impurity problem and its solution to the challenging problems posed by the recent work on heavy fermions and the high temperature superconductors. The first eight chapters cover, largely in chronological order, the techniques which have been developed to deal with single impurity problems. Some of these techniques, such as Green's functions, Feynman diagrams and perturbation theory, are covered in standard many-body texts (for example, Fetter and Walecka, 1971: Abrikosov, Gorkov and Dzyaloshinski, 1975: Mahan, 1990). Others may be less familiar so for these techniques I have included general introductory sections in the relevant chapters, and some appendices with further details, in order to make the text as self-contained as possible. The aim has been to make the book readable at two levels. At the higher level I have tried to present the general development of ideas, the emphasis being on the results of the theory and the general physical picture that emerges. The equations at this level are included to make it clear how these results are obtained. I have tried to make it readable also at a second more detailed level by including enough information in the text and appendices so that one can work from one equation to the next.