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The authors present a theory to describe spin transport across a polymer sandwiched between magnetic contacts and propose organic spin devices based on this theory. It is found that even a weak magnetic field can significantly modify spin transport in polymers through spin precession. This sensitivity can be exploited to design ultrasensitive magnetometers and low-power magnetic-field-effect transistors. It is shown that, at room temperature, the organic magnetometers are capable of detecting sub-nanotesla magnetic fields, and the I-V characteristics of the magnetic-field-effect transistors can be strongly modified by magnetic fields of a few gauss with response times of a few nanoseconds.