  Type Ia supernovae are associated with thermonuclear explosions of white
dwarf stars. Combustion processes convert material in nuclear reactions and
release the energy required to explode the stars. At the same time, they
produce the radioactive species that power radiation and give rise to the
formation of the observables. Therefore, the physical mechanism of the
combustion processes, as reviewed here, is the key to understand these
astrophysical events. Theory establishes two distinct modes of propagation for
combustion fronts: subsonic deflagrations and supersonic detonations. Both are
assumed to play an important role in thermonuclear supernovae. The physical
nature and theoretical models of deflagrations and detonations are discussed
together with numerical implementations. A particular challenge arises due to
the wide range of spatial scales involved in these phenomena. Neither the
combustion waves nor their interaction with fluid flow and instabilities can be
directly resolved in simulations. Substantial modeling effort is required to
consistently capture such effects and the corresponding techniques are
discussed in detail. They form the basis of modern multidimensional
hydrodynamical simulations of thermonuclear supernova explosions. The problem
of deflagration-to-detonation transitions in thermonuclear supernova explosions
is briefly mentioned.
