In this paper, the pure strategy subgame perfect equilibria of a general class of stopping time games are studied. It is shown that there always exists a natural class of Markov Perfect Equilibria, called stopping equilibria. Such equilibria can be computed as a solution of a single agent stopping time problem, rather than of a fixed point problem. A complete characterization of stopping equilibria is presented. Conditions are given under which the outcomes of such equilibria span the set of all possible outcomes from perfect equilibria. Two economic applications of the theory, product innovations and the timing of asset sales, are discussed.