In this paper, the multiagent pursuit-evasion (MPE) games are solved in order to obtain optimal strategic policies for all players. In these games, multiple pursuers attempt to intercept multiple evaders who try to avoid capture. A graph-theoretic approach is employed to study the interactions of the agents with limited sensing capabilities, such that distributed control policies are obtained for every agent. Furthermore, the minimization of performance indices associated with the goals of the agents is guaranteed. Nash equilibrium among the players is obtained by means of optimal policies that use the solutions of the Hamilton-Jacobi-Isaacs (HJI) equations of the game. Minmax strategies are also proposed to guarantee a security-level performance when the solutions of the HJI equations for Nash equilibrium do not exist. Scenarios for finite-time capture and for asymptotic rendezvous are analyzed, and emergent behaviors are obtained by means of modifications of the proposed general-case performance indices. The containment control results are shown to be special cases of the solutions of the MPE games.