The application of Type Ia supernovae (SNe Ia) as distance indicators in cosmology calls for a sound understanding of these objects. Recent years have seen a brisk development of astrophysical models which explain SNe Ia as thermonuclear explosions of white dwarf stars. While the evolution of the progenitor is still uncertain, the explosion mechanism certainly involves the propagation of a thermonuclear flame through the white dwarf star. Three-dimensional hydrodynamical simulations allowed to study a wide variety of possibilities involving subsonic flame propagation (deflagrations), flames accelerated by turbulence, and supersonic detonations. These possibilities lead to a variety of scenarios. I review the currently discussed approaches and present some recent results from simulations of the turbulent deflagration model and the delayed detonation model.