Climatic warming of about 0.5 ° C in the global mean since the 1970s has strongly increased the occurrence-probability of heat extremes on monthly to seasonal time scales. For the 21st century, climate models predict more substantial warming. Here we show that the multi-model mean of the CMIP5 (Coupled Model Intercomparison Project) climate models accurately reproduces the evolution over time and spatial patterns of the historically observed increase in monthly heat extremes. For the near-term (i.e., by 2040), the models predict a robust, several-fold increase in the frequency of such heat extremes, irrespective of the emission scenario. However, mitigation can strongly reduce the number of heat extremes by the second half of the 21st century. Unmitigated climate change causes most (>50%) continental regions to move to a new climatic regime with the coldest summer months by the end of the century substantially hotter than the hottest experienced today. We show that the land fraction experiencing extreme heat as a function of global mean temperature follows a simple cumulative distribution function, which depends only on natural variability and the level of spatial heterogeneity in the warming.