We derive in this work a local nonlinear small-gain theorem in the framework of input-to-state stability for discrete time systems. Our primary objective is to show that, as in the continuous-time context, these discrete-time nonlinear small-gain theorems are very effective in stability analysis and synthesis for various classes of discrete-time control systems. Two converse Lyapunov theorems for discrete exponential stability are developed to assist these applications. New results in stability and stabilization presented in this paper are significant extensions of previous work by other authors (IEEE Trans. Automat. Control 38 (993) 1398; 39 (1994) 2340; 33 (1988) 1082). (C) 2004 Elsevier Ltd. All rights reserved.