Changes in the intensity or frequency of extreme climate events can profoundly increase the disruption caused by climate change1,2,3,4. The more extreme these events, the greater the potential to push ecosystems and communities beyond their ability to cope3,5. The rate at which existing high temperature records have been broken has increased in response to rising global greenhouse gas emissions (GGHGEs)2,6,7,8, and the rate at which historical records are surpassed is projected to increase further over the coming century1,2,9,10. Here we examine future events that will be so extreme that they will not have been experienced previously. Record setting in 22 climate models11 indicates that, by the end of the twenty-first century, under business-as-usual increases in GGHGEs (representative concentration pathway (RCP) 8.5 (ref. 12)), high monthly mean temperature records will be set in approximately 58% of the world every year, and in 67% of least developed countries and 68% of small island developing states. These figures all drop to 14% under a scenario with much lower GHG concentrations (RCP2.6 (ref. 12)). In any given year, the likelihood of ‘smashing’ at least one monthly record by more than 1.0 °C is much less likely under RCP2.6 than it is under RCP8.5 (1.1 versus 8.9%).