Many continuous-time control laws are shown to achieve finite-time convergence. However, when discretized, such property is, most of the times, lost. In this article we study the conditions guaranteeing the finite-time convergence of discrete-time mappings and also investigate the cases when such maps can be interpreted as the backward-Euler discretization of system with a maximal monotone right-hand side. Finite-time stability is characterized. Many examples illustrate the theoretical developments.