We investigate a new property of computing systems called weak stabilization. Although this property is strictly weaker than the well-known property of stabilization, weak stabilization is superior to stabilization in several respects. In particular, adding delays to a system preserves the system property of weak stabilization, but does not necessarily preserve its stabilization property. Because most implementations are bound to add arbitrary delays to the systems being implemented, weakly stabilizing systems are much easier to implement than stabilizing systems. We also prove the following important result. A weakly stabilizing system that has a finite number of states is in fact stabilizing assuming that the system execution is strongly fair. Finally, we discuss an interesting method for composing several weakly stabilizing systems into a single weakly stabilizing system.