This book is an inquiry into the mathematical structure of the theory of singular perturbation. The book can be studied in various ways, depending on the readers interests and motivation. For example, the practitioner of heuristic analysis who is mainly interested in applications may read superficially the first three chapters, study mainly Chapters 4 and 5 and dismiss the rest. However, if he would care to do some reading in Chapter 6. he may discover ways of turning the heuristic analysis of some specific problem into a rigorous theory. On the other hand, the mathematician interested in the techniques of proof of the validity of formal approximations may concentrate on Chapter 6, while a student of elliptic p.d.e.'s can content himself with parts of Chapter 6 and study Chapter 7. Let me further mention that the first three chapters are rather self-contained, and should be of interest to those who are puzzled and fascinated (as I have been for many years) by the somewhat bizarre foundations of the method of matched expansions.