Some basic results of supervisory control theory are extended to the setting of omega-languages, formal languages consisting of infinite strings. The extension permits the investigation of both liveness and safety issues in the control of discrete-event systems. A new controllability property appropriate to the infinitary setting (omega-controllability) is defined; this language property captures in a natural way the limitations of available control actions. It is shown that every specification language contains a unique maximal omega-controllable sublanguage, representing the least upper bound of the set of achievable closed-loop sublanguages. This supremal omega-controllable sublanguage allows a simple formulation of necessary and sufficient conditions for the solvability of an infinitary supervisory control problem. The problems of effectively deciding solvability of the control problem and of effectively synthesizing appropriate supervisors are solved for the case where the plant is represented by a deterministic Buchi automaton and the specification of legal behavior by a deterministic Rabin automaton.