Abstract. In this paper we prove the existence of ^-equilibrium stationary strategies for non-zero-sum stochastic games when the reward functions and transitions satisfy certain separability conditions. We also prove some results for positive and discounted zero-sum stochastic games when the state space is infinite.Introduction. A stochastic game is determined by five objects: S, A, B, q, r. Here S is a nonempty Borel subset of a Polish space, the set of states of the system. A is a nonempty Borel subset of a Polish space, the set of actions available to player I; B is the set of actions for player II. The law of motion q associates Borel measurably with each is, a, b) E S X A X B a probability measure on the Borel subsets of S. Let r¡(s, a, b), i = 1, 2, be the reward functions for I and II, respectively, when s is the state and a, b are the actions of I and IL As a consequence of the actions chosen by the players, two things happen: players I and II receive rxis, a, b), r2is, a, b) and the system moves to a new state s' according to qi-\s, a, b). Then the whole process is repeated from the new state s'. The problem is to find whether they have suitable Nash equilibrium strategies.A strategy II for I is a sequence (II,, TL2, . . .) where Hn specifies the action to be chosen on the Aith day depending on the past history. A strategy n is called stationary if there is a Borel map /: S -> PA (the class of all probability