  To constrain the nature of the very first stars, we investigate the collapse
and fragmentation of primordial, metal-free gas clouds. We explore the physics
of primordial star formation by means of three-dimensional simulations of the
dark matter and gas components, using smoothed particle hydrodynamics, under a
wide range of initial conditions, including the initial spin, the total mass of
the halo, the redshift of virialization, the power spectrum of the DM
fluctuations, the presence of HD cooling, and the number of particles employed
in the simulation. We find characteristic values for the temperature, T ~ a few
100 K, and the density, n ~ 10^3-10^4 cm^-3, characterising the gas at the end
of the initial free-fall phase. These values are rather insensitive to the
initial conditions. The corresponding Jeans mass is M_J ~ 10^3 M_sun. The
existence of these characteristic values has a robust explanation in the
microphysics of H2 cooling, connected to the minimum temperature that can be
reached with the H2 coolant, and to the critical density at which the
transition takes place betweeb levels being populated according to NLTE, and
according to LTE.
  In all cases, the gas dissipatively settles into an irregular, central
configuration which has a filamentary and knotty appearance. The fluid regions
with the highest densities are the first to undergo runaway collapse due to
gravitational instability, and to form clumps with initial masses ~ 10^3 M_sun,
close to the characteristic Jeans scale. These results suggest that the first
stars might have been quite massive, possibly even very massive with M_star >
100 M_sun.
