We examine contemporaneous perfect epsilon-equilibria, in which a player's actions after every history, evaluated at the point of deviation from the equilibrium, must be within epsilon of a best response. This concept implies, but is stronger than, Radner's ex ante perfect epsilon-equilibrium. A strategy profile is a contemporaneous perfect epsilon-equilibrium of a game if it is a subgame perfect equilibrium in a perturbed game with nearly the same payoffs, with the converse holding for pure equilibria.