Social dilemmas are central to human society. Depletion of natural resources, climate protection, security of energy supply, and workplace collaborations are all issues that give rise to social dilemmas. Since cooperative behaviour in a social dilemma is individually costly, Nash equilibrium predicts that humans should not cooperate. Yet experimental studies show that people do cooperate even in anonymous one shot situations. However, in spite of the large number of participants in many modern social dilemmas, little is known about the effect of group size on cooperation. Does larger group size favour or prevent cooperation? We address this problem both experimentally and theoretically. Experimentally, we have found that there is no general answer: it depends on the strategic situation. Specifically, we have conducted two experiments, one on a one shot Public Goods Game and one on a one shot N person Prisoner Dilemma. We have found that larger group size favours the emergence of cooperation in the Public Goods game, but prevents it in the Prisoner dilemma. On the theoretical side, we have shown that this behaviour is not consistent with either the Fehr and Schmidt model or (a one parameter version of) the Charness and Rabin model. Looking for models explaining our findings, we have extended the cooperative equilibrium model from two player social dilemmas to some N person social dilemmas and we have shown that it indeed predicts the above mentioned regularities. Since the cooperative equilibrium is parameter free, we have also made a direct comparison between its predictions and experimental data. We have found that the predictions are neither strikingly close nor dramatically far from the experimental data.