Our high-time-resolution observations reveal that individual main pulses from the Crab pulsar contain one or more short-lived microbursts. Both the energy and duration of bursts measured above 1 GHz can vary dramatically in less than a millisecond. These fluctuations are too rapid to be caused by propagation through turbulence in the Crab Nebula or the interstellar medium; they must be intrinsic to the radio emission process in the pulsar. The mean duration of a burst varies with frequency as $ u^{-2}$, significantly different from the broadening caused by interstellar scattering. We compare the properties of the bursts to some simple models of microstructure in the radio emission region.