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 mega–electron volts to 10 giga–electron volts) flares observed by the AGILE satellite in September 2010 and October 2007. In both cases, the total gamma-ray flux increased by a factor of three compared with the non-flaring flux. The flare luminosity and short time scale favor an origin near the pulsar, and we discuss Chandra Observatory x-ray and Hubble Space Telescope 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 an approximately 1-day time scale.