Evaluation of potential crop yields is important for global food security assessment because it represents the biophysical 'ceiling' determined by variety, climate and ambient CO2. Statistical approaches have limitations when assessing future potential yields, while large differences between results obtained using process-based models reflect uncertainties in model parameterisations. Here we simulate the potential yield of wheat across the present-day wheat-growing areas, using a new global model that couples a parameter-sparse, optimality-based representation of gross primary production (GPP) to empirical functions relating GPP, biomass production and yield. The model reconciles the transparency and parsimony of statistical models with a mechanistic grounding in the standard model of C3 photosynthesis, and seamlessly integrates photosynthetic acclimation and CO2 fertilization effects. The model accurately predicted the CO2 response observed in FACE experiments, and captured the magnitude and spatial pattern of EARTHSTAT 'attainable yield' data in 2000 CE better than process-based models in ISIMIP. Global simulations of potential yield during 1981–2016 were analysed in parallel with global historical data on actual yield, in order to test the hypothesis that environmental effects on modelled potential yields would also be shown in observed actual yields. Higher temperatures are thereby shown to have negatively affected (potential and actual) yields over much of the world. Greater solar radiation is associated with higher yields in humid regions, but lower yields in semi-arid regions. Greater precipitation is associated with higher yields in semi-arid regions. The effect of rising CO2 is reflected in increasing actual yield, but trends in actual yield are stronger than the CO2 effect in many regions, presumably because they also include effects of crop breeding and improved management. We present this hybrid modelling approach as a useful addition to the toolkit for assessing global environmental change impacts on the growth and yield of arable crops.