In this work we study the input-to-state stability (ISS) property for discrete-time nonlinear systems. It is shown that most Iss results for continuous-time nonlinear systems in the current literature can be extended to the discrete-time case. Several equivalent characterizations of ISS are introduced and two ISS small-gain theorems are proved for nonlinear and interconnected discrete-time systems. ISS stabilizability is discussed and comparisons with the continuous-time case are made. As in the continuous time framework, where the notion Iss found wide applications, we expect that this notion will provide a useful tool in areas related to stability and stabilization for nonlinear discrete time systems as well.