We inter-compare the mathematical formulation of ten models of terrestrial net primary productivity (NPP) and their functional responses to temperature (T), carbon dioxide (CO2), soil water (W) and photosynthetically active radiation (PAR). The models span a broad spectrum of complexity of approaches from the original, empirical Miami model, through β-factor, and global-average box models, to the dynamic global vegetation models (DGVMs) TRIFFID and BIOME3. Five of the models separate photosynthesis and respiration, although only three directly consider biochemistry. Equations for all the models are given in a complete, compact, standardized format. NPP responses to temperature differ markedly: β-factor models show only a tiny increase with temperature, empirical models predict a modest S-shaped response saturating above 35 °C (a surrogate for drought effects), biochemical and quasi-biochemical models show a peaked response with an optimum in the range 15–25 °C except for DEMETER (<10 °C). Qualitative differences in whether respiration exhibits a peaked or exponential response to temperature have only a modest effect on NPP. The CO2 response of NPP is qualitatively similar in all but one model that misrepresents it. Where a water stress response of NPP is represented, it is saturating in all but the same model, where NPP declines under high soil moisture. Where represented, the PAR response of NPP is saturating, but the saturation point differs considerably. When responses to two environmental variables are combined, the second variable typically just scales the response to the first. In biochemical models, the temperature optimum of NPP increases with CO2, and it also increases noticeably with PAR in TRIFFID and a derivative of it. Where different plant functional types (PFTs) are represented, C4 grasses have the greatest maximum NPP at a higher optimum temperature than any C3 PFT. Boreal, needle-leaf trees generally have the lowest NPP. Additional uncertainties in modelling respiration, stomatal conductance, light-use efficiency, and scaling to the canopy level are discussed. We suggest that a concerted effort to standardize definitions, notation, units and PFTs would add a degree of transparency that is currently lacking in NPP modelling.