Climate models with increased levels of carbon dioxide predict that global warming causes heating in the tropics, but investigations of ancient climates based on palaeodata have generally indicated cool tropical temperatures during supposed greenhouse episodes. For example, in the Late Cretaceous and Eocene epochs there is abundant geological evidence for warm, mostly ice-free poles, but tropical sea surface temperatures are generally estimated to be only 15-23°C, based on oxygen isotope palaeothermometry of surface-dwelling planktonic foraminifer shells. Here we question the validity of most such data on the grounds of poor preservation and diagenetic alteration. We present new data from exceptionally well preserved foraminifer shells extracted from impermeable clay-rich sediments, which indicate that for the intervals studied, tropical sea surface temperatures were at least 28-32°C. These warm temperatures are more in line with our understanding of the geographical distributions of temperature-sensitive fossil organisms and the results of climate models with increased CO2 levels.