We analyze the long-run behaviour of stochastic dynamics in well-mixed populations and in spatial games with local interactions. We review results concerning the effect of the number of players and the noise level on the stochastic stability of Nash equilibria. To address the problem of equilibrium selection in spatial games with many players, we introduce a concept of ensemble stability. The standard stochastic stability describes a long-run behaviour of systems with a fixed number of players in the zero-noise limit. On the contrary, the ensemble stability is concerned with a fixed (but nevertheless low) noise level in the limit of the infinite number of players. We present examples of games in which when the number of players increases or the noise level decreases, a population undergoes a transition between its equilibria. In particular, it may happen that a risk-dominant and Pareto-efficient strategy, which is stochastically stable, in the long run is played with an arbitrarily small probability if the noise level is low and the number of players is big enough.