The method of matched asymptotic expansions (MAE) is well suited to so-called boundary layer problems, where the solution varies rapidly near boundaries of their domain of definition and smoothly elsewhere. The chapter starts by explaining how these distinct behaviours can be separately approximated and then joined to make uniformly valid composite approximations. Following the general pedagogical approach of the book, the technique is introduced on an elementary example that can be solved exactly, so that the workings and the efficiency of the method can easily be illustrated. Matched asymptotic expansions are then applied in detail to the SIR epidemiological model in the limit of large reproduction number, elasticas, laser switch-on transients, and other problems, each time to illustrate subtleties of the method or difficulties that can be encountered in realistic research problems. Here, the emphasis is on the first approximation, for which the implementation of the method is lightest. The question of dealing with higher-order corrections, notably using Van Dyke’s matching rule, is therefore deferred to the end of the chapter.