To be sustainable, the extractive process of fishing requires biomass renewal via primary production driven by solar energy. Primary production required (PPR) estimates how much primary production is needed to replace the biomass of fisheries landings removed from marine ecosystems. Here, we examine the historical fishing behaviour of global fishing fleets, which parts of the food web they rely on, which ecosystems they fish and how intensively. Highly mobile European and Asian fleets have moved to ever more distant productive waters since the 1970s, especially once they are faced with the costs of access agreements for exclusive economic zones (EEZs) declared by host countries. We examine fleet PPR demands in the context of large marine ecosystems (LMEs), which are frequently fished with PPR demands well above their average primary productivity (PP). In some cases, this was mitigated by subsequent emigration of fleets or by management intervention. Fleet movements, however, have stressed additional marine areas, including the EEZs of developing countries. This suggests the potential for spatial serial depletion, if fishing capacity is not reduced to more sustainable PP removal levels. Fundamentally, fishing is limited by solar-powered PP limits. Fishing beyond solar production has occurred, but in the future, marine systems may not be as forgiving, especially if overfishing and climate change compromise their resilience.