Darwinian evolution has to provide an explanation for cooperative behaviour. Theories of cooperation are based on kin selection (dependent on genetic relatedness), group selection and reciprocal altruism. The idea of reciprocal altruism usually involves direct reciprocity: repeated encounters between the same individuals allow for the return of an altruistic act by the recipient. Here we present a new theoretical framework, which is based on indirect reciprocity and does not require the same two individuals ever to meet again. Individual selection can nevertheless favour cooperative strategies directed towards recipients that have helped others in the past. Cooperation pays because it confers the image of a valuable community member to the cooperating individual. We present computer simulations and analytic models that specify the conditions required for evolutionary stability of indirect reciprocity. We show that the probability of knowing the 'image' of the recipient must exceed the cost-to-benefit ratio of the altruistic act. We propose that the emergence of indirect reciprocity was a decisive step for the evolution of human societies.