Despite major advances in climate science over the last 30 years, persistent uncertainties in projections of future climate change remain. Climate projections are produced with increasingly complex models that attempt to represent key processes in the Earth system, including atmospheric and oceanic circulations, convection, clouds, snow, sea ice, vegetation, and interactions with the carbon cycle. Uncertainties in the representation of these processes feed through into a range of projections from the many state-of-the-art climate models now being developed and used worldwide. For example, despite major improvements in climate models, the range of equilibrium global warming due to doubling carbon dioxide still spans a range of more than 3. Here a promising way to make use of the ensemble of climate models to reduce the uncertainties in the sensitivities of the real climate system is reviewed. The emergent constraint approach uses the model ensemble to identify a relationship between an uncertain aspect of the future climate and an observable variation or trend in the contemporary climate. This review summarizes previous published work on emergent constraints and discusses the promise and potential dangers of the approach. Most importantly, it argues that emergent constraints should be based on well-founded physical principles such as the fluctuation-dissipation theorem. This review will stimulate physicists to contribute to the rapidly developing field of emergent constraints on climate projections, bringing to it much-needed rigor and physical insights.