<p>This paper studies the problem of finite-time stability (FTS) for nonlinear discrete-time systems with delayed impulses. Based on impulsive control and Lyapunov stability theory, we present some new FTS theorems for these systems under stabilizing impulses and destabilizing impulses, including sufficient conditions and settling-time estimations. It can be observed that the settling-time of discrete-time systems with delayed impulses depends not only on the initial state but also on the type of impulses and the delay magnitude. As compared with the impulse-free case, a smaller settling-time bound can be obtained when a finite-time stable system is subject to stabilizing impulses without delay. A small delay increases the settling-time bound but keeps it smaller than the impulse-free case, while a large delay leads to an even larger bound. Conversely, under impulsive disturbance, a larger settling-time bound can be derived regardless of delays. Finally, numerical examples are provided to illustrate the effectiveness of the obtained theoretical results.</p>