Quitting games are sequential games in which, at any stage, each player has the choice between continuing and quitting. The game ends as soon as at least player chooses to quit; player i then receives a payoff r, which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is 0. We prove the existence of cyclic E-equilibrium under some assumptions on the payoff function (r sub s). We prove on an example that our result is essentially optimal. We also discuss the relation to Dynkin's stopping games, and provide a generalization of our result to these games.