  We present results from the first cosmological simulations which study the
onset of primordial, metal-free (population III), cosmic star formation and the
transition to the present-day, metal-rich star formation (population II-I),
including molecular (H$_2$, HD, etc.) evolution, tracing the injection of
metals by supernov{\ae} into the surrounding intergalactic medium and following
the change in the initial stellar mass function (IMF) according to the
metallicity of the corresponding stellar population. Our investigation
addresses the role of a wide variety of parameters (critical metallicity for
the transition, IMF slope and range, SN/pair-instability SN metal yields, star
formation threshold, resolution, etc.) on the metal-enrichment history and the
associated transition in the star formation mode. All simulations present
common trends. Metal enrichment is very patchy, with rare, unpolluted regions
surviving at all redshifts, inducing the simultaneous presence of metal-free
and metal-rich star formation regimes. As a result of the rapid pollution
within high-density regions due to the first SN/pair-instability SN, local
metallicity is quickly boosted above the critical metallicity for the
transition. The population III regime lasts for a very short period during the
first stages of star formation ($\sim 10^7\,\rm yr$), and its average
contribution to the total star formation rate density drops rapidly below $\sim
10^{-3}-10^{-2}$.
