The necessary and sufficient conditions for the completion of a class of pursuit— evasion games are given. The state variables of the pursuer are assumed to be defined by a set of linear differential equations and the evader's state variables are given by a vector valued non-linear differential equation. The game is defined to be completed if certain (or all) state variables of the pursuer ; i.e. position, velocity, etc. are equal to the corresponding state variables of the evader. The Krein's L-method is used to determine the existence of the optimal completion time and the optimal player strategies. The general results are specialized to the case in which the evader's state vector is also governed by a set of linear differential equations. A computational technique—steepest descent method has been developed to obtain the numerical solution of the game. Three examples are presented to illustrate the method. The first one deals with a linear, second order, pursuit-evasion game with constant coefficients in which completion occurs when the position component of the player's state vector coincide. A second linear example is selected to illustrate the effectiveness with which the approach developed in this paper can be used to obtain a sub-optimal solution even if the exact solution cannot be achieved. In the last example, a non-linear pursuit–evasion game is considered. Here it is shown that the technique developed may be applied to a certain class of non-linear pursuit–evasion game although the procedures are much more involved.