  Improving our understanding of the initial conditions and earliest stages of
star formation is crucial to gain insight into the origin of stellar masses,
multiple systems, and protoplanetary disks. We review the properties of
low-mass dense cores as derived from recent millimeter/submillimeter
observations of nearby molecular clouds and discuss them in the context of
various contemporary scenarios for cloud core formation and evolution. None of
the extreme scenarios can explain all observations. Pure laminar ambipolar
diffusion has relatively long growth times for typical ionization levels and
has difficulty satisfying core lifetime constraints. Purely hydrodynamic
pictures have trouble accounting for the inefficiency of core formation and the
detailed velocity structure of individual cores. A possible favorable scenario
is a mixed model involving gravitational fragmentation of turbulent molecular
clouds close to magnetic criticality. The evolution of the magnetic field and
angular momentum in individual cloud cores after the onset of gravitational
collapse is also discussed. In particular, we stress the importance of
radiation-magnetohydrodynamical processes and resistive MHD effects during the
protostellar phase. We also emphasize the role of the formation of the
short-lived first (protostellar) core in providing a chance for
sub-fragmentation into binary systems and triggering MHD outflows. Future
submillimeter facilities such as Herschel and ALMA will soon provide major new
observational constraints in this field. On the theoretical side, an important
challenge for the future will be to link the formation of molecular clouds and
prestellar cores in a coherent picture.
