High temperatures are detrimental to crop yields and could lead to global warming-driven reductions in agricultural productivity. To assess future threats, the majority of studies used process-based crop models, but their ability to represent effects of high temperature has been questioned. Here we show that an ensemble of nine crop models reproduces the observed average temperature responses of US maize, soybean and wheat yields. Each day >30 °C diminishes maize and soybean yields by up to 6% under rainfed conditions. Declines observed in irrigated areas, or simulated assuming full irrigation, are weak. This supports the hypothesis that water stress induced by high temperatures causes the decline. For wheat a negative response to high temperature is neither observed nor simulated under historical conditions, since critical temperatures are rarely exceeded during the growing season. In the future, yields are modelled to decline for all three crops at temperatures >30 °C. Elevated CO₂can only weakly reduce these yield losses, in contrast to irrigation.

France is a major crop producer, with a production share of approx. 20% within the European Union. Yet, a discussion has recently started whether French yields are stagnating. While for wheat previous results are unanimously pointing to recent stagnation, there is contradictory evidence for maize and few to no results for other crops. Here we analyse a data set with more than 120,000 yield observations from 1900 to 2016 for ten crops (barley, durum and soft wheat, maize, oats, potatoes, rapeseed, sugar beet, sunflower and wine) in the 96 mainland French départements (NUTS3 administrative division). We dissect the evolution of yield trends over time and space, analyse yield variation and evaluate whether growth of yields has stalled in recent years. Yields have, on average across crops, multiplied four-fold over the course of the 20 ^th century. While absolute yield variability has increased, the variation relative to the mean has halved – mean yields have increased faster than their variability. But growth of yields has stagnated since the 1990’s for winter wheat, barley, oats, durum wheat, sunflower and wine on at least 25% of their areas. Reaching yield potentials is unlikely as a cause for stagnation. Maize, in contrast, shows no evidence for stagnation.