The Axiom of Determinacy is associated with a game of perfect information, in which two players take turns choosing natural numbers. Associated with the game is a rule defining the conditions under which each player wins. If one player has a winning strategy, the game is said to be determined; the Axiom of Determinacy states that all such games are determined. The axiom has important consequences for the real numbers: it implies that every set of real numbers is Lebesgue measurable and has the Baire property. The axiom is also intimately related to large cardinals. For example, it implies the existence of an inner model of L(R) with infinitely many Woodin cardinals. The talk will begin with a rigorous definition of the Axiom of Determinacy. Some elementary consequences of the axiom will then be discussed. The talk should be interesting to both logicians and non-logicians.