Large-scale, three-dimensional numerical simulations of the deflagration stage of a thermonuclear supernova explosion show the formation and evolution of a highly convoluted turbulent flame in the gravitational field of an expanding carbon-oxygen white dwarf. The flame dynamics are dominated by the gravity-induced Rayleigh-Taylor instability that controls the burning rate. The thermonuclear deflagration releases enough energy to produce a healthy explosion. The turbulent flame, however, leaves large amounts of unburned and partially burned material near the star center, whereas observations that imply these materials are present only in outer layers. This disagreement could be resolved if the deflagration triggers a detonation.