  Observations now probe the star formation history of the Universe back to a
redshift of $z\sim5$. We investigate whether the predictions of semi-analytic
models of galaxy formation based on hierarchical Cold Dark Matter (CDM) type
models are in agreement with these direct observations and also with the
``fossil'' evidence contained in constraints on the ages of present day
early-type galaxies. Previous models predicted that the star formation rate
density falls off rather steeply at $z\ga 2$, and correspondingly that the
majority of the stars in the Universe formed at relatively low redshift. We
investigate the effect of including a bursting mode of star formation, assuming
that galaxy-galaxy mergers trigger starbursts and using the merger rate that
arises naturally in the CDM merging hierarchy. The resulting starbursts
substantially increase the global star formation rate at high redshift, leading
to predictions that are in good agreement with the star formation rate density
at $z\sim3$ obtained from sub-millimeter observations (SCUBA) and optical/UV
estimates after correction for dust extinction. The mass of stars formed at $z
\ge 3$ is correspondingly in better agreement with the fossil evidence. We also
investigate complementary global quantities such as the mass of cold gas and
the average metallicity of cold gas as a function of redshift, and the
integrated extra-galactic background light. We argue that these observations,
taken together, provide strong constraints on the star formation history of the
Universe, and that hierarchical models of the CDM type are in reasonable
agreement with these observations when starbursts are included.
