Inflation is a paradigm for the physics of the early universe, receiving a great deal of experimental support. Essentially all inflationary models lead to a regime called eternal inflation, a situation where inflation never ends globally, but it only ends locally. The resulting picture is that of a multiverse: an infinite number of "bubbles", each of them containing an infinite universe. Extracting meaningful predictions on what physics we should expect to observe within our bubble encounters severe ambiguities. Hence the need to adopt a prescription to regulate the diverging spacetime volume of the multiverse. The problem of defining such a prescription is the measure problem, which is a major challenge of modern theoretical cosmology. This book provides an introduction to the measure problem, and proposes and explores a promising candidate solution: the scale-factor cutoff measure. This book is intendend for experts of the field as well as for graduate students willing to be initiated to this fascinating subject.