Kurt Gödel's famous First Incompleteness Theorem shows that, for any sufficiently rich theory that contains enough arithmetic, there are some arithmetical truths the theory can express but cannot prove. How is this remarkable result established? This short book explains. It also discusses Gödel's Second Incompleteness Theorem. The aim is to make the Theorems available, clearly and accessibly, even to those with a quite limited formal background.The first edition was based on much-downloaded lecture notes for a course given in Cambridge for many years. This second edition is expanded and extensively revised.