Discrete-event processes are modelled by state-machines in the Ramadge-Wonham framework with control by a feedback event disablement mechanism. In this paper concepts of stabilization of discrete-event processes are defined and investigated. We examine the possibility of driving a process (under control) from arbitrary initial states to a prescribed subset of the state set and then keeping it there indefinitely. This stabilization property is studied also with respect to ‘open-loop’ processes (i.e. uncontrolled processes) and their asymptotic behaviour is characterized. To this end, such well known classical concepts of dynamics as invariant sets and attractors are redefined and characterized in the discrete-event control framework. We provide polynomial time algorithms for verifying various types of attraction and for the. synthesis of attractors.