  Turbulent fragmentation determines where and when protostellar cores form,
and how they contract and grow in mass from the surrounding cloud material.
This process is investigated, using numerical models of self-gravitating
molecular cloud dynamics. Molecular cloud regions without turbulent driving
sources, or where turbulence is driven on large scales, exhibit rapid and
efficient star formation in a clustered mode, whereas interstellar turbulence
that carries most energy on small scales results in isolated star formation
with low efficiency. The clump mass spectrum of shock-generated density
fluctuations in pure hydrodynamic, supersonic turbulence is not well fit by a
power law, and it is too steep at the high-mass end to be in agreement with the
observational data. When gravity is included in the turbulence models, local
collapse occurs, and the spectrum extends towards larger masses as clumps merge
together, a power-law description dN/dM ~ M^nu becomes possible with slope nu <
-2. In the case of pure gravitational contraction, i.e. in regions without
turbulent support, the clump mass spectrum is shallower with nu = -3/2. The
mass spectrum of protostellar cores in regions without turbulent support and
where turbulence is replenished on large-scales, however, is well described by
a log-normal or by multiple power laws, similar to the stellar IMF at low and
intermediate masses. In the case of small-scale turbulence, the core mass
spectrum is too flat compared to the IMF for all masses.
