A large discrepancy exists between the dire impacts that most natural scientists project we could face from climate change and the modest estimates of damages calculated by mainstream economists. Economic assessments of climate change risks are intended to be comprehensive, covering the full range of physical impacts and their associated market and non-market costs, considering the greater vulnerability of poor people and the challenges of adaptation. Available estimates still fall significantly short of this goal, but alternative approaches that have been proposed attempt to address these gaps. This review seeks to provide a common basis for natural scientists, social scientists, and modellers to understand the research challenges involved in evaluating the economic risks of climate change. Focusing on the estimation processes embedded in economic integrated assessment models and the concerns raised in the literature, we summarise the frontiers of research relevant to improving quantitative damage estimates, representing the full complexity of the associated systems, and evaluating the impact of the various economic assumptions used to manage this complexity.