We study systems in which the equilibrium point varies discontinuously according to a well defined state or time-dependent switching law. We refer to those systems as systems with switching equilibria. To motivate our study, we describe a class of problems in engineering and biology that can be formulated using such systems. We study stability and convergence properties of those systems under various switching rules. In particular we prove convergence under arbitrary switching, time-dependent and state-dependent switching laws. In the time-dependent switching case we highlight connections between the relaxation theorem corresponding to differential inclusions, Pulse-Width-Modulation (PWM) and averaging theory.