P. Padoan (ICREA & Institute of Cosmos Sciences, University of Barcelona, Spain),
C. Federrath (School of Mathematical Sciences, Monash University, Australia),
G. Chabrier (Ecole Normale Superieure de Lyon, France),
N. Evans (The University of Texas at Austin, United States),
D. Johnstone (Herzberg Institute of Astrophysics, National Research Council of Canada),
J. Jorgensen (Star and Planet Formation Center, University of Copenhagen, Denmark),
C. McKee (University of California, Berkeley, United States),
A. Nordlund (Niels Bohr Institute & Star and Planet Formation Center, University of Copenhagen)

We review recent advances in the analytical and numerical modeling of the star formation rate in molecular clouds and discuss the available observational constraints. We focus on molecular clouds as the fundamental star formation sites, rather than on the larger-scale processes that form the clouds and set their properties. Molecular clouds are shaped into a complex filamentary structure by supersonic turbulence, with only a small fraction of the cloud mass channeled into collapsing protostars over a free-fall time of the system. In recent years, the physics of supersonic turbulence has been widely explored with computer simulations, leading to statistical models of this fragmentation process, and to the prediction of the star formation rate as a function of fundamental physical parameters of molecular clouds, such as the virial parameter, the rms Mach number, the compressive fraction of the turbulence forcing, and the ratio of magnetic to gas pressure. Infrared space telescopes, as well as ground-based observatories have provided unprecedented probes of the filamentary structure of molecular clouds and the location of forming stars within them. We may thus be on the verge of a breakthrough in our understanding of star formation, and particularly of the star formation rate of molecular clouds.

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