This paper develops new recursive methods for studying stationary sequential equilibria in games with private monitoring. We first consider games where play has occurred forever into the past and develop methods for analyzing a large class of stationary strategies, where the main restriction is that the strategy can be represented as a finite automaton. For a subset of this class, strategies which depend only on the players’ signals in the last _k_ periods, these methods allow the construction of _all_ pure strategy equilibria. We then show that each sequential equilibrium in a game with infinite histories defines a correlated equilibrium for a game with a start date and derive simple necessary and sufficient conditions for determining if an _arbitrary_ correlation device yields a correlated equilibrium. This allows, for games with a start date, the construction of all pure strategy _sequential_ equilibria in this subclass.