  In this paper, we review the present state of theoretical models of
thermonuclear supernovae, and compare their predicitions with the constraints
derived from observations of Type Ia supernovae. The diversity of explosion
mechanisms usually found in one-dimensional simulations is a direct consequence
of the impossibility to resolve the flame structure under the assumption of
spherical symmetry. Spherically symmetric models have been successful in
explaining many of the observational features of Type Ia supernovae, but they
rely on two kinds of empirical models: one that describes the behaviour of the
flame on the scales unresolved by the code, and another that takes account of
the evolution of the flame shape. In contrast, three-dimensional simulations
are able to compute the flame shape in a self-consistent way, but they still
need a model for the propagation of the flame in the scales unresolved by the
code. Furthermore, in three dimensions the number of degrees of freedom of the
initial configuration of the white dwarf at runaway is much larger than in one
dimension. Recent simulations have shown that the sensitivity of the explosion
output to the initial conditions can be extremely large. New paradigms of
thermonuclear supernovae have emerged from this situation, as the Pulsating
Reverse Detonation. The resolution of all these issues must rely on the
predictions of observational properties of the models, and their comparison
with current Type Ia supernova data, including X-ray spectra of Type Ia
supernova remnants.
