We introduce the Virgo Consortium's Evolution and Assembly of GaLaxies and their Environments (EAGLE) project, a suite of hydrodynamical simulations that follow the formation of galaxies and supermassive black holes in cosmologically representative volumes of a standard Λ cold dark matter universe. We discuss the limitations of such simulations in light of their finite resolution and poorly constrained subgrid physics, and how these affect their predictive power. One major improvement is our treatment of feedback from massive stars and active galactic nuclei (AGN) in which thermal energy is injected into the gas without the need to turn off cooling or decouple hydrodynamical forces, allowing winds to develop without predetermined speed or mass loading factors. Because the feedback efficiencies cannot be predicted from first principles, we calibrate them to the present-day galaxy stellar mass function and the amplitude of the galaxy-central black hole mass relation, also taking galaxy sizes into account. The observed galaxy stellar mass function is reproduced to ≲ 0.2 dex over the full resolved mass range, 108 < M*/M⊙ ≲ 1011, a level of agreement close to that attained by semi-analytic models, and unprecedented for hydrodynamical simulations. We compare our results to a representative set of low-redshift observables not considered in the calibration, and find good agreement with the observed galaxy specific star formation rates, passive fractions, Tully-Fisher relation, total stellar luminosities of galaxy clusters, and column density distributions of intergalactic C IV and O VI. While the mass-metallicity relations for gas and stars are consistent with observations for M* ≳ 109 M⊙ (M* ≳ 1010 M⊙ at intermediate resolution), they are insufficiently steep at lower masses. For the reference model, the gas fractions and temperatures are too high for clusters of galaxies, but for galaxy groups these discrepancies can be resolved by adopting a higher heating temperature in the subgrid prescription for AGN feedback. The EAGLE simulation suite, which also includes physics variations and higher resolution zoomed-in volumes described elsewhere, constitutes a valuable new resource for studies of galaxy formation.