In this talk, we introduce Schmidt's game, which in recent years has been used extensively to prove theorems in number theory. We will contrast Schmidt's game with the Banach-Mazur game to motivate investigation into certain set theoretic properties of the game. The central concept of this talk will be the property of a game that one of the two players has a winning strategy (determinacy). Under certain strong hypotheses, we present a proof that the Banach-Mazur game is determined (which has implications for subsets of the real line, for instance). In light of this proof, we will examine how Schmidt's game differs and sketch a proof of a partial result of the determinacy of Schmidt's game.