Persistent divergences among the predictions of complex carbon-cycle models include differences in the sign as well as the magnitude of the response of global terrestrial primary production to climate change. Such problems with current models indicate an urgent need to reassess the principles underlying the environmental controls of primary production. The global patterns of annual and maximum monthly terrestrial gross primary production (GPP) by C3 plants are explored here using a simple first-principles model based on the light-use efficiency formalism and the Farquhar model for C3 photosynthesis. The model is driven by incident photosynthetically active radiation (PAR) and remotely sensed green-vegetation cover, with additional constraints imposed by low-temperature inhibition and CO2 limitation. The ratio of leaf-internal to ambient CO2 concentration in the model responds to growing-season mean temperature, atmospheric dryness (indexed by the cumulative water deficit, Δ E) and elevation, based on an optimality theory. The greatest annual GPP is predicted for tropical moist forests, but the maximum (summer) monthly GPP can be as high, or higher, in boreal or temperate forests. These findings are supported by a new analysis of CO2 flux measurements. The explanation is simply based on the seasonal and latitudinal distribution of PAR combined with the physiology of photosynthesis. By successively imposing biophysical constraints, it is shown that partial vegetation cover - driven primarily by water shortage - represents the largest constraint on global GPP.