Ocean temperature extremes such as marine heatwaves are expected to intensify in coming decades due to anthropogenic global warming. Reported ecological and economic impacts of marine heatwaves include coral bleaching, local extinction of mangrove and kelp forests, and elevated mortalities of invertebrates, fishes, seabirds, and marine mammals. In contrast, little is known about the impacts of marine heatwaves on microbes that regulate biogeochemical processes in the ocean. Here we analyse the physical and biogeochemical output of a near-global eddy-resolving ocean circulation model (OFAM3) simulation to elucidate the impacts of marine heatwaves on phytoplankton blooms in the tropics and temperate oceans. The model successfully simulates many of the observed marine heatwave events in recent decades. Our model-based analysis reveals contrasting responses of phytoplankton blooms to marine heatwave events which are dependent on the background nutrient concentration. In nutrient replete conditions, marine heatwaves are associated with elevated phytoplankton blooms, whereas in nutrient limited conditions, they are associated with reduced blooms. We analyse the daily-mean satellite observations of sea surface temperatures, chlorophyll a concentrations, and carbon biomass to assess the validity of the findings of the model-based analysis. Lastly, we discuss the implications of these findings for marine ecosystems and biogeochemistry under future climate.