  We give an overview of the current understanding of Type Ia supernovae
relevant for their use as cosmological distance indicators. We present the
physical basis to understand their homogeneity of the observed light curves and
spectra and the observed correlations. SNe Ia have been well established as
distance indicators on the 10 % level. However, the quest for the nature of the
dark energy requires improvements in the accuracy to the 2 to 3 % level, we
must understand the diversity within the SNe Ia population, and its evolution
with redshift. Based on detailed models for the progenitors, explosions, light
curves and spectra, we discuss signatures of thermonuclear explosions, and the
implications for cosmology. We emphasize the relation between LC properties and
spectra because, for local SNe~Ia, the diversity becomes apparent the
combination of spectra and LCs whereas, by enlarge, we have to for high-z
objects. At some examples, we show how we can actually probe the properties of
the progenitor, its environment, and details of the explosion physics.

Astrophysics