Discrete-Event processes are modeled 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 behavior 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. Finally, we provide polynomial time algorithms for verifying various types of attraction and for the synthesis of attractors.