In John Nash?s proofs for the existence of (Nash) equilibria based on Brouwer?s theorem, an iteration mapping is used. A continuous?time analogue of the same mapping has been studied even earlier by Brown and von Neumann. This differential equation has recently been suggested as a plausible boundedly rational learning process in games. In the current paper we study this Brown?von Neumann?Nash dynamics for the case of continuous strategy spaces. We show that for continuous payoff functions, the set of rest points of the dynamics coincides with the set of Nash equilibria of the underlying game. We also study the asymptotic stability properties of rest points. While strict Nash equilibria may be unstable, we identify sufficient conditions for local and global asymptotic stability which use concepts developed in evolutionary game theory.