Non-human primates are marked by well-developed prosocial and cooperative tendencies as reflected in the way they support each other in fights, hunt together, share food and console victims of aggression. The proximate motivation behind such behaviour is not to be confused with the ultimate reasons for its evolution. Even if a behaviour is ultimately self-serving, the motivation behind it may be genuinely unselfish. A sharp distinction needs to be drawn, therefore, between (i) altruistic and cooperative behaviour with knowable benefits to the actor, which may lead actors aware of these benefits to seek them by acting cooperatively or altruistically and (ii) altruistic behaviour that offers the actor no knowable rewards. The latter is the case if return benefits occur too unpredictably, too distantly in time or are of an indirect nature, such as increased inclusive fitness. The second category of behaviour can be explained only by assuming an altruistic impulse, which-as in humans-may be born from empathy with the recipient's need, pain or distress. Empathy, a proximate mechanism for prosocial behaviour that makes one individual share another's emotional state, is biased the way one would predict from evolutionary theories of cooperation (i.e. by kinship, social closeness and reciprocation). There is increasing evidence in non-human primates (and other mammals) for this proximate mechanism as well as for the unselfish, spontaneous nature of the resulting prosocial tendencies. This paper further reviews observational and experimental evidence for the reciprocity mechanisms that underlie cooperation among non-relatives, for inequity aversion as a constraint on cooperation and on the way defection is dealt with.