We study repeated games in which players observe a public outcome that imperfectly signals the actions played. We provide conditions guaranteeing that any feasible, individually rational payoff vector of the stage game can arise as a perfect equilibrium of the repeated game with sufficiently little discounting. The central condition requires that there exist action profiles with the property that, for any two players, no two deviations--one by each player--give rise to the same probability distribution over public outcomes. The results apply to principal-agent, partnership, oligopoly, and mechanism-design models, and to one-shot games with transferable utilities.