To explain the evolution of cooperation by natural selection has been a major goal of biologists since Darwin. Cooperators help others at a cost to themselves, while defectors receive the benefits of altruism without providing any help in return. The standard game dynamical formulation is the 'Prisoner's Dilemma', in which two players have a choice between cooperation and defection. In the repeated game, cooperators using direct reciprocity cannot be exploited by defectors, but it is unclear how such cooperators can arise in the first place. In general, defectors are stable against invasion by cooperators. This understanding is based on traditional concepts of evolutionary stability and dynamics in infinite populations. Here we study evolutionary game dynamics in finite populations. We show that a single cooperator using a strategy like 'tit-for-tat' can invade a population of defectors with a probability that corresponds to a net selective advantage. We specify the conditions required for natural selection to favour the emergence of cooperation and define evolutionary stability in finite populations.