International cooperation to tackle complex common resource problems like climate change is extremely difficult. Although there is broad agreement on the nature of the problem and what is required to solve it, many nations continue to block any meaningful action for solution. This global cooperation crisis is baffling in the light of recent evidence about the surprisingly cooperative disposition of human beings. Research from social and natural sciences points to an unmistakable conclusion: people cooperate all the time, and they enjoy doing so. This picture of human behaviour is at odds with common assumptions about people being narrowly—and exclusively—self-interested, and prompts the question that we address in this paper: why, if we are so good cooperating at interpersonal levels, is international cooperation so hard? We address this question in three steps. First, we review the recent multidisciplinary evidence demonstrating that people cooperate much more than rational-theory models predict, and that this might stem from a natural, evolved, predisposition to cooperate. Second, we argue that there are seven basic mechanisms that determine whether or not cooperation is successful or sustainable: reciprocity, trust, communication, reputation, fairness, enforcement and we-identity. We group these mechanisms in a ‘cooperation hexagon’ that summarizes the current consensus about what makes cooperation work. Finally, we discuss what these findings mean for global cooperation. We argue that power games are not enough to explain off current international cooperation blockades. A new, comprehensive theory of international cooperation must be compatible with the recent insights about the fundamentally cooperative nature of human behaviour. We suggest that the search for this theory be made in three directions: a) establish how cooperation scales up from interpersonal to lager scales, and how the basic mechanisms of cooperation behave under conditions of unprecedented complexity and rapid change; b) investigate cooperation at the ‘meso-level’ of global governance—the relatively small group of people who represent nations in international discussions and institutions—a key interface between interpersonal and inter-institutional motivations for cooperation; and c) examine patterns of international cooperation in the light of the cooperation hexagon, to ascertain whether international cooperation blockades are the result of the underprovisioning of the basic mechanisms of cooperation, and how these mechanisms can be used as criteria for designing better institutions for global governance.