Conventions form an essential part of human social and cultural behaviour and may also be important to other animal societies. Yet, despite the wealth of evidence that has accumulated for culture in non-human animals, we know surprisingly little about non-human conventions beyond a few rare examples. We follow the literature in behavioural ecology and evolution and define conventions as systematic behaviours that solve a coordination problem in which two or more individuals need to display complementary behaviour to obtain a mutually beneficial outcome. We start by discussing the literature on conventions in non-human primates from this perspective and conclude that all the ingredients for conventions to emerge are present and therefore that they ought to be more frequently observed. We then probe emergence of conventions by using a unique novel experimental system in which pairs of Guinea baboons (Papio papio) can voluntarily participate together in touch-screen based cognitive testing and we show that conventions readily emerge in our experimental setup and that they share three fundamental properties of human conventions (arbitrariness, stability and efficiency). These results question the idea that observational learning, and imitation in particular, is necessary to establish conventions, they suggest that positive reinforcement is enough.