Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players’ behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems. We apply this theory, and in particular the concepts of chain recurrence, attractors, and the Conley index, to prove a general impossibility result: There exist games for which any dynamics will fail to converge, from certain initial conditions, to the set of Nash equilibria. The games which help prove this impossibility result are degenerate, but we conjecture that an analogous result holds, under complexity assumptions, for nondegenerate games. We also prove a stronger result for approximate Nash equilibria: For a set of games of positive measure, there are no game dynamics that converge to the set of approximate Nash equilibria for some substantial approximation bound. These impossibility results also apply to dynamics with memory. We argue that these results further weaken the appeal of the Nash equilibrium as the solution concept of choice in game theory, and discuss alternatives suggested by the dynamics point of view.