Kurt G del's famous First Incompleteness Theorem shows that for any sufficiently rich theory that contains enough arithmetic, there are some arithmetical truths the theory cannot prove. How is this remarkable result proved? This short book explains. It also discusses G del's Second Incompleteness Theorem. Based on lecture notes for a course given in Cambridge for many years, the aim is to make the Theorems available, clearly and accessibly, even to those with a quite limited formal background.