Magnetic fields are believed to play an important role in the evolution of molecular clouds, from their large scale structure to dense cores, protostellar envelopes, and protoplanetary disks. How important is unclear, and whether magnetic fields are the dominant force driving star formation at any scale is also unclear. In this review we examine the observational data which address these questions, with particular emphasis on high angular resolution observations. Unfortunately the data do not clarify the situation. It is clear that the fields are important, but to what degree we dont yet know. Observations to date have been limited by the sensitivity of available telescopes and instrumentation. In the future ALMA and the SKA in particular should provide great advances in observational studies of magnetic fields, and we discuss which observations are most desirable when they become available.
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