  We introduce the Virgo Consortium's EAGLE project, a suite of hydrodynamical
simulations that follow the formation of galaxies and black holes in
representative volumes. 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 AGN in which thermal energy is injected into
the gas without the need to turn off cooling or 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 z~0 galaxy stellar mass function and the amplitude of the
galaxy-central black hole mass relation, also taking galaxy sizes into account.
The observed galaxy mass function is reproduced to $\lesssim 0.2$ dex over the
full mass range, $10^8 < M_*/M_\odot \lesssim 10^{11}$, 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 CIV and OVI. While
the mass-metallicity relations for gas and stars are consistent with
observations for $M_* \gtrsim 10^9 M_\odot$, they are insufficiently steep at
lower masses. The gas fractions and temperatures are too high for clusters of
galaxies, but for groups these discrepancies can be resolved by adopting a
higher heating temperature in the subgrid prescription for AGN feedback. EAGLE
constitutes a valuable new resource for studies of galaxy formation.
