Abstract : A young and energetic pulsar powers the well-known Crab Nebula. Here we describe two separate gamma-ray (photon energy >100 MeV) flares from this source detected by the Large Area Telescope on board the Fermi Gamma-ray Space Telescope. The first flare occurred in February 2009 and lasted approximately 16 days. The second flare was detected in September 2010 and lasted approximately 4 days. During these outbursts the gamma-ray flux from the nebula increased by factors of four and six, respectively. The brevity of the flares implies that the gamma rays were emitted via synchrotron radiation from PeV (10^15 eV) electrons in a region smaller than 1.4 10^-2 pc. These are the highest energy particles that can be associated with a discrete astronomical source, and they pose challenges to particle acceleration theory.