The problem of preserving stability properties under small perturbations for the solutions of difference equations is considered. The approach used is to study the behavior of the solutions of the perturbed difference equation with respect to the solutions of the original unperturbed difference equations. This leads to the introduction of notions which parallel the usual concepts of stability, asymptotic stability, instability and the like for the behavior of the perturbed solutions with respect to the unperturbed ones.The principal technique employed is an extension of Lyapunov’s direct method based on the difference of the two solutions. A series of theorems is obtained yielding criteria for each type of behavior for the perturbed solutions in terms of the existence of a discrete Lyapunov-type function with appropriate properties.