A pursuit problem and an evasion problem are formulated and results are obtained for the case in which the dynamics are governed by linear differential equations and the terminal set is a linear manifold in the state space. Conditions are given ensuring the existence of an open set in the phase space such that if the initial state belongs to this set, termination of the pursuit game can be achieved. Conditions are also given ensuring that for any initial state not in the terminal manifold the evasion game can be prolonged indefinitely.