  We provide a status report on our current understanding of the mass scales
for Pop III.1 and Pop III.2 stars. Since the last review (Norman 2008),
substantial progress has been made both numerically and analytically on the
late stages of protostellar cloud core collapse, protostar formation and
accretion, and stellar evolution taking into account cloud core properties and
radiative feedback effects. Based on this, there are growing indications that
primordial stars forming from purely cosmological initial conditions (Pop
III.1) were substantially more massive than stars forming in preionized gas
(Pop III.2) where HD cooling is important. Different stellar endpoints are
predicted for these two types of Pop III stars with different chemical
enrichment signatures: the former die as pair instability supernovae or
intermediate mass black holes, whereas the latter die as iron core-collapse
supernovae, leaving behind neutron star and stellar black hole remnants. We
review recent simulations which show evidence for binary fragmentation at high
densities, and comment on the significance of these results. We then summarize
an attempt to directly calculate the Pop III.1 IMF taking into account the
latest numerical and analytical models. We conclude with suggestions for the
kind of simulations needed next to continue improving our understanding of Pop
III star formation, which is a necessary input to understanding high redshift
galaxy formation.
