  A key question for supernova cosmology is whether the peak luminosities of
Type Ia supernovae (SNe Ia) are sufficiently free from the effects of cosmic
and galactic evolution. To answer this question, we review the currently
popular scenario of SN Ia progenitors, i.e., the single degenerate scenario for
the Chandrasekhar mass white dwarf (WD) models. We identify the progenitor's
evolution with two channels: (1) the WD+RG (red-giant) and (2) the WD+MS (near
main-sequence He-rich star) channels. The strong wind from accreting WDs plays
a key role, which yields important age and metallicity effects on the
evolution.
  We suggest that the variation of the carbon mass fraction $X$(C) in the C+O
WD (or the variation of the initial WD mass) causes the diversity of SN Ia
brightness. This model can explain the observed dependence of SNe Ia brightness
on the galaxy types. We then predict how SN Ia brightness evolves along the
redshift (with changing metallicity and age) for elliptical and spiral
galaxies. Such evolutionary effects along the redshift can be corrected as has
been made for local SNe Ia.
  We also touch on several related issues: (1) the abundance pattern of stars
in dwarf spheroidal galaxies in relation to the metallicity effect on SNe Ia,
(2) effects of angular momentum brought into the WD in relation to the
diversities and the fate of double degenerates, and (3) possible presence of
helium in the peculiar SN Ia 2000cx in relation to the sub-Chandrasekhar mass
model.
