We study zero-sum repeated games where the minimizing player has to pay a certain cost each time he changes his action. We show that the value of the game exists in stationary strategies, which depend solely on the previous action of the player (and not the entire history), and we provide a full characterization of the value and the optimal strategies. The strategies exhibit a robustness property and typically do not change with a small perturbation of the switching costs. We also consider a case where the player is limited to playing completely history-independent strategies. Naturally, this limitation worsens his situation. We deduce a bound on his loss in the general case as well as more precise bounds when more assumptions regarding the game are introduced.