The well known Crab Nebula is at the center of the SN1054 supernova remnant. It consists of a rotationally-powered pulsar interacting with a surrounding nebula through a relativistic particle wind. The emissions originating from the pulsar and nebula have been considered to be essentially stable. Here we report the detection of strong gamma-ray (100 MeV-10 GeV) flares observed by the AGILE satellite in September, 2010 and October, 2007. In both cases, the unpulsed flux increased by a factor of 3 compared to the non-flaring flux. The flare luminosity and short timescale favor an origin near the pulsar, and we discuss Chandra Observatory X-ray and HST optical follow-up observations of the nebula. Our observations challenge standard models of nebular emission and require power-law acceleration by shock-driven plasma wave turbulence within a ~1-day timescale.