  We introduce the Illustris Project, a series of large-scale hydrodynamical
simulations of galaxy formation. The highest resolution simulation,
Illustris-1, covers a volume of $(106.5\,{\rm Mpc})^3$, has a dark mass
resolution of ${6.26 \times 10^{6}\,{\rm M}_\odot}$, and an initial baryonic
matter mass resolution of ${1.26 \times 10^{6}\,{\rm M}_\odot}$. At $z=0$
gravitational forces are softened on scales of $710\,{\rm pc}$, and the
smallest hydrodynamical gas cells have an extent of $48\,{\rm pc}$. We follow
the dynamical evolution of $2\times 1820^3$ resolution elements and in addition
passively evolve $1820^3$ Monte Carlo tracer particles reaching a total
particle count of more than $18$ billion. The galaxy formation model includes:
primordial and metal-line cooling with self-shielding corrections, stellar
evolution, stellar feedback, gas recycling, chemical enrichment, supermassive
black hole growth, and feedback from active galactic nuclei. At $z=0$ our
simulation volume contains about $40,000$ well-resolved galaxies covering a
diverse range of morphologies and colours including early-type, late-type and
irregular galaxies. The simulation reproduces reasonably well the cosmic star
formation rate density, the galaxy luminosity function, and baryon conversion
efficiency at $z=0$. It also qualitatively captures the impact of galaxy
environment on the red fractions of galaxies. The internal velocity structure
of selected well-resolved disk galaxies obeys the stellar and baryonic
Tully-Fisher relation together with flat circular velocity curves. In the
well-resolved regime the simulation reproduces the observed mix of early-type
and late-type galaxies. Our model predicts a halo mass dependent impact of
baryonic effects on the halo mass function and the masses of haloes caused by
feedback from supernova and active galactic nuclei.
