Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets $A \subseteq \mathbf{R}$ is the game determined? ???? Rules: The two players alternate moves starting with player I. Each move $a_n$ is legal iff it is a real number and $0 < a_n$, and for $n > 1, a_n < a_{n - 1}$. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff $\sum a_n$ exists and $\sum a_n \in A$. We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinancy and the determinancy of other well-known and much-studied games.