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), characterizing 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 solar mass. The existence of these characteristic values has a robust explanation in the microphysics of H_2 cooling, connected to the minimum temperature that can be reached with the H_2 coolant, and to the critical density at which the transition takes place between levels being populated according to non-LTE (NLTE), and according to LTE. In all cases, the gas dissipatively settles into an irregular, central configuration that 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 solar mass, close to the characteristic Jeans scale. These results suggest that the first stars might have been quite massive, possibly even very massive with M_* approx> 100 solar mass. After a gas element has undergone runaway collapse, and has reached densities in excess of 10~8 cm~(-3), a sink particle is created. This procedure allows us to follow the evolution of the overall system beyond the point where the first nonlinear region would otherwise force the calculation to a halt. These later evolutionary stages, during which the clumps grow in mass due to accretion and merging with other clumps, are quite sensitive to the initial conditions. The key process in building up very massive clumps, with masses up to a few times 10~4 solar mass, is merging between clumps. Since the merging rate sensitively depends on the density of the gas, halos with the highest degree of central concentration are able to assemble the most massive clumps. Among these are halos with a low spin (λ approx= 0.01), and with DM fluctuations imprinted according to a white-noise spectrum.
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