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 \simeq 1 - 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 \sim 10 ^ { 12 } ~ { } \mathrm { M } _ { \odot } 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 .