  We study low-metallicity star formation with a set of high-resolution
hydrodynamics simulations for various gas metallicities over a wide range
$0$--$10^{-3} \ {\rm Z}_{\bigodot}$. Our simulations follow non-equilibrium
chemistry and radiative cooling by adopting realistic elemental abundance and
dust size distribution. We examine the condition for the fragmentation of
collapsing clouds (cloud fragmentation; CF) and of accretion discs (disc
fragmentation; DF). We find that CF is suppressed due to rapid gas heating
accompanied with molecular hydrogen formation even with efficient dust cooling
for metallicities $\gtrsim 10^{-5} \ {\rm Z}_{\bigodot}$. Instead, DF occurs in
almost all runs regardless of metallicity. We also find that, in the accretion
discs, the growth of the protostellar systems is overall oligarchic. The
primary protostar grows through the accretion of gas, and secondary protostars
form through the interaction of spiral arms or the break-up of a rapidly
rotating protostar. Despite vigorous fragmentation, a large fraction of
secondary protostars are destroyed through mergers or tidal disruption events.
For a few hundred years after the first adiabatic core formation, only several
protostars survive in the disc, and the total mass of protostars is
$0.52$--$3.8 \ {\rm M}_{\bigodot}$.
