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 ~ 50%. Carbon exported out of the upper ocean also fuels the productivity of the mesopelagic zone, including significant fisheries. 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 warming 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 and future projections of export, as well as identifying the observations and model developments required to achieve more robust characterisation of this important planetary carbon flux. We identify particle fragmentation and zooplankton vertical migration as the mechanisms most likely to substantially influence the magnitude of present-day modelled export flux. Of the processes sufficiently understood to allow implementation in climate models, projections of future export flux and feedbacks to climate are likely to be most sensitive to changes in phytoplankton and particle size spectra, and to temperature-dependent remineralisation. “Known unknown” processes which are not currently represented in models and will have an uncertain impact on future projections include particle stickiness and fish vertical migration. With the advent of new observational technologies, such as biogeochemical-Argo floats and miniaturised camera systems, we will be able to better parameterize models and thus decrease uncertainties in current and future export flux.