We investigate the evolution of galaxy masses and star formation rates in the Evolution and Assembly of Galaxies and their Environment (EAGLE) simulations. These comprise a suite of hydrodynamical simulations in a Λ cold dark matter cosmogony with subgrid models for radiative cooling, star formation, stellar mass-loss and feedback from stars and accreting black holes. The subgrid feedback was calibrated to reproduce the observed present-day galaxy stellar mass function and galaxy sizes. Here, we demonstrate that the simulations reproduce the observed growth of the stellar mass density to within 20 per cent. The simulations also track the observed evolution of the galaxy stellar mass function out to redshift z = 7, with differences comparable to the plausible uncertainties in the interpretation of the data. Just as with observed galaxies, the specific star formation rates of simulated galaxies are bimodal, with distinct star forming and passive sequences. The specific star formation rates of star-forming galaxies are typically 0.2 to 0.5 dex lower than observed, but the evolution of the rates track the observations closely. The unprecedented level of agreement between simulation and data across cosmic time makes EAGLE a powerful resource to understand the physical processes that govern galaxy formation.