The export flux of organic carbon from the upper ocean is the starting point of the transfer and long term storage of photosynthetically-fixed carbon in the deep ocean. This “biological carbon pump” is a significant component of the global carbon cycle, reducing atmospheric CO2 levels by ~ 200 ppm. Carbon exported out of the upper ocean also fuels the productivity of the mesopelagic zone, including significant fisheries. Here we show that, despite its importance, export flux is poorly constrained in Earth System Models, with the modelled range in projected future global-mean changes due to climate change spanning +1.8 to -41%. Fundamental constraints to understanding export flux arise because a myriad of interconnected processes make the biological carbon pump challenging to both observe and model. Our synthesis prioritises the processes likely to be most important to include in modern-day estimates (particle fragmentation and zooplankton vertical migration) and future projections (phytoplankton and particle size spectra, and temperature-dependent remineralisation) of export. We also identify the observations required to achieve more robust characterisation, and hence improved model parameterization, of export flux, and thus decrease uncertainties in current and future estimates of this important planetary carbon flux.