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We study the evolutionary game dynamics of a two-strategy game. In infinite populations, the well-known replicator equations describe the deterministic evolutionary dynamics. There are three generic selection scenarios. The dynamics of a finite group of players has received little attention. We provide a framework for studying stochastic evolutionary game dynamics in finite populations. We define a Moran process with frequency dependent fitness. We find that there are eight selection scenarios. And for a given payoff matrix, a number of these sceanrios can occur for different population size. Our results have interesting applications in biology and economics. In particular, we obtain new results on the evolution of cooperation in the classic repeated Prisoner's Dilemma game. We show that a single cooperator using a reciprocal strategy such as Tit-For-Tat can invade a population of defectors with a probability that corresponds to a net selective advantage. We specify the conditions for natural selection to favor the emergence of cooperation and derive conditions for evolutionary stability in finite populations.