The gamma-ray space telescopes Agile and Fermi reported bright day-long flares in the Crab Nebula above 100 MeV energy photons, presumably of synchrotron origin. This discovery implies that electrons and positrons are accelerated to PeV energies in a milliGauss magnetic field, challenging the most established models of particle acceleration and pulsar wind nebulae. In this talk, I will argue that the flares could be powered by magnetic reconnection in the nebula. Relativistic test-particle simulations show that the particles are naturally focused into a thin fan beam, and accelerated deep inside the reconnection layer. I will show that this scenario provides a viable explanation for the gamma-ray flares in the Crab Nebula.