This paper presents a self-triggered MPC controller design strategy for linear systems with state and input constraints. Based on the so-called relaxed dynamic programming inequality, the synthesis procedure determines both the updated MPC control action and the next triggering time. The resulting self-triggered MPC control law preserves stability and constraint satisfaction and also satis es a certain speci fied performance requirement without requiring stabilizing terminal constraints. A robust self-triggered MPC scheme, based on the tube-MPC idea, is also presented for linear systems with persistent bounded additive disturbances. Simulation examples illustrate the e ffectiveness of our proposed self-triggered MPC scheme.