Integrated Assessment Models (IAMs) have gained a prominent role in the climate science-policy interface. The article reconstructs the evolution of IAMs and their changing role in this interface, investigating how and why IAMs have become so prominent. Based on literature analysis, quantitative document analysis and semi-structured interviews, we describe the historic evolution of the interactions between IAMs and policy-making between 1970 and 2015. We identify five historic phases in which IAMs played distinct mediating roles between science and policy, succeeding to adjust their scenario efforts to the continuously changing demands for knowledge from the policy community. In explaining the prominent role of IAMs, we differentiate between background conditions (material and sociological) and more contextual factors, most notably the flexible, hybrid and broad nature of IAMs as well as the pro-active character of the IAM community to enhance their policy relevance. We draw on the notion of institutional work to explain this success. In light of the urgency of responding to the climate crisis, we suggest that the IAM community may expand their scope of anticipated futures and consider engaging a wider range of publics and societal stakeholders beyond the science-policy interface.