© The Author(s), 2017. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Surveys in Geophysics 38 (2017): 1529–1568, doi:10.1007/s10712-017-9428-0.Trade-wind cumuli constitute the cloud type with the highest frequency of occurrence on Earth, and it has been shown that their sensitivity to changing environmental conditions will critically influence the magnitude and pace of future global warming. Research over the last decade has pointed out the importance of the interplay between clouds, convection and circulation in controling this sensitivity. Numerical models represent this interplay in diverse ways, which translates into different responses of trade-cumuli to...