In this paper, event-triggered strategies for control of discrete-time systems are proposed and analyzed. Similarly to the continuous-time case, the plant is assumed input-to-state stable with respect to measurement errors and the control law is updated once a triggering condition involving the norm of a measurement error is violated. The results are also extended to a self-triggered formulation, where the next control updates are decided at the previous ones, thus relaxing the need for continuous monitoring of the measurement error. The overall framework is then used in a novel Model Predictive Control approach. The results are illustrated through simulated examples.