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.