  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.
