  Understanding the physical processes that drive star formation is a key
challenge for galaxy formation models. In this article we study the tight
correlation between the star formation rate (SFR) and stellar mass of galaxies
at a given redshift, how halo growth influences star formation, and star
formation histories of individual galaxies. We study these topics using
Illustris, a state-of-the-art cosmological hydrodynamical simulation of galaxy
formation. Illustris reproduces the observed relation (the star formation main
sequence; SFMS) between SFR and stellar mass at redshifts z=0 and z=4, but at
intermediate redshifts of z~2, the simulated SFMS has a significantly lower
normalisation than reported by observations. The scatter in the relation is
consistent with the observed scatter. However, the fraction of outliers above
the SFR-stellar mass relation in Illustris is less than that observed. Galaxies
with halo masses of ~10^{12} solar masses dominate the SFR density of the
Universe, in agreement with the results of abundance matching. Furthermore,
more-massive galaxies tend to form the bulk of their stars at high redshift,
which indicates that `downsizing' occurs in Illustris. We also studied the star
formation histories of individual galaxies, including the use of a principal
component analysis decomposition. We find that for fixed stellar mass, galaxies
that form earlier have more-massive black holes at z=0, indicating that star
formation and black hole growth are tightly linked processes in Illustris.
While many of the properties of normal star-forming galaxies are
well-reproduced in the Illustris simulation, forming a realistic population of
starbursts will likely require higher resolution and probably a more
sophisticated treatment of star formation and feedback from stars and black
holes.
