Three aspects of the avoidance control problem are considered : (i) avoidance from a particular initial state ; (ii) global avoidance ; and (iii) avoidance from all initial states belonging to a subset of the state space. Necessary and sufficient conditions for the existence of an avoidance control are presented for situations (i) to (iii) above. The criteria are in terms of a finite-dimensional optimization problem and are suit able for numerical computation.