It has long been observed, as a basic arithmetic phenomenon, that if two numbers with no common factors are each divisible by relatively large high powers of integers, then their sum tends not to be itself divisible by a large high power. This phenomenon is borne out by some celebrated theorems (e.g., Mordell's Conjecture, finiteness of the number of solutions to the Catalan problem, Fermat's Last Theorem). Two decades ago it was given quantitative precision in the form of the ABC Conjecture of Masser and Oesterlé. In my three lectures I will try to give motivation for this conjecture, to discuss the further arithmetic problems it inspires, and to show how central is the role it now plays in Arithmetic.