This paper introduces a class of contest models in which each player decides when to stop a privately observed Brownian motion with drift and incurs costs depending on his stopping time. The player who stops his process at the highest value wins a prize. We prove existence and uniqueness of a Nash equilibrium outcome and derive the equilibrium distribution in closed form. As the variance tends to zero, the equilibrium outcome converges to the symmetric equilibrium of an all-pay auction. For two players and constant costs, each player’s equilibrium profit decreases if the drift increases, the variance decreases, or the costs decrease.