The standard model of repeated games assumes perfect synchronization in the timing of decisions between the players. In many natural settings, however, choices are made asynchronously so that only one player can move at any given time. This paper studies a family of repeated settings in which choices are asynchronous. Initially, we examine, as a canonical model, a simple two person alternating move game of pure coordination. There, it is shown that for sufficiently patient players, there is a unique perfect equilibrium payoff which Pareto dominates all other payoffs. The result generalizes to any finite number of players and any game in a class of asynchonrously repeated games which includes both stochastic and deterministic repetition. The results complement a recent Folk Theorem by Dutta (1995) for stochastic games which can be applied to asynchronously repeated games if a full dimensionality condition holds. A critical feature of the model is the inertia in decisions. We show how the inertia in asynchronous decisions determines the set of equilibrium payoffs.