The response of marine plankton to climate change is of critical importance to the oceanic food web and fish stocks. We use a 60-year ocean basin-wide data set comprising > 148,000 samples to reveal huge differences in range changes associated with climate change across 35 plankton taxa. While the range of dinoflagellates and copepods tended to closely track the velocity of climate change (the rate of isotherm movement), the range of the diatoms moved much more slowly. Differences in range shifts were up to 900 km in a recent warming period, with average velocities of range movement between 7 km per decade northwards for taxa exhibiting niche plasticity and 99 km per decade for taxa exhibiting niche conservatism. The differing responses of taxa to global warming will cause spatial restructuring of the plankton ecosystem with likely consequences for grazing pressures on phytoplankton and hence for biogeochemical cycling, higher trophic levels and biodiversity.