Among hamadryas baboons, Papio hamadryas, but not savanna baboons, P. anubis, P. cynocephalus, P. ursinus, the most frequent interaction observed among males, especially adults, in the context of rivalry over females is greeting. This paper investigates its possible functions. Interactions of greeting (asymmetrical, symmetrical and unreciprocated), aggression and coalitions observed between 20 males, members of a large multi-harem colony of baboons (hamadryas, cynocephalus and their hybrids) housed in the Madrid Zoo, were recorded. Males in possession of sexually mature females, and especially those in control of oestrous females (possessors), tended to initiate and receive symmetrical greetings, tended not to be greeted back when they initiated a greeting and not to greet back when they received a greeting, and tended to be frequent targets of coalition solicitations. Greetings were always given by possessors to rivals. It is suggested that, in the context of rivalry over females, greeting could be a strategy of negotiation whereby possessors and their rivals can assess the situation, influence each other's roles in the relationship, and eventually resolve conflicts without having to fight. Qualitative evidence concerning developmental careers and contexts are also used to substantiate this interpretation. Differences between male hamadryas and savanna baboons in the use of strategies directed towards rivals and possessors in the context of competition over females are also discussed. © 1991 The Association for the Study of Animal Behaviour.