By its elegant inclusion of slow and fast scales in a single ansatz, the method of multiple scales is particularly appealing to physicists. Its use is first demonstrated on a weakly damped oscillator before being described in fuller detail. Several equivalent implementations exist and are contrasted: two-time vs. more-than-two-time. Central to the method is the solvability condition, which is explained in several manners: it can be found either by demanding that some “secular terms” be excluded from the approximate solution or by invoking the so-called “Fredholm alternative”. Next to linearly damped oscillations, the weakly non-linear Van der Pol oscillator is described in detail. This and other examples are used to show how to apply the method using computer algebra systems such as Mathematica. Other applications are reviewed in detail, notably delayed-differential equations and weakly non-linear wave packets governed by fully vectorial Maxwell equations. It is also shown how the WKB method can be viewed as a particular application of the present method. Finally, an alternative and somewhat equivalent method, namely the renormalisation group method, is presented and explained.