  For carbon-oxygen white dwarfs accreting hydrogen or helium at rates in the
range ~1-10 x 10^(-8) Msun/y, a variety of explosive outcomes is possible well
before the star reaches the Chandrasekhar mass. These outcomes are surveyed for
a range of white dwarf masses (0.7 - 1.1 Msun), accretion rates (1 - 7 x
10^(-8) Msun/y), and initial white dwarf temperatures (0.01 and 1 Lsun). The
results are particularly sensitive to the convection that goes on during the
last few minutes before the explosion. Unless this convection maintains a
shallow temperature gradient, and unless the density is sufficiently high, the
accreted helium does not detonate. Below a critical helium ignition density,
which we estimate to be 5 - 10 x 10^5 g cm^(-3), either helium novae or helium
deflagrations result. The hydrodynamics, nucleosynthesis, light curves, and
spectra of a representative sample of detonating and deflagrating models are
explored. Some can be quite faint indeed, powered at peak for a few days by the
decay of 48Cr and 48V. Only the hottest, most massive white dwarfs considered
with the smallest helium layers, show reasonable agreement with the light
curves and spectra of common Type Ia supernovae. For the other models,
especially those involving lighter white dwarfs, the helium shell mass exceeds
0.05 Msun and the mass of the 56Ni that is synthesized exceeds 0.01 Msun. These
explosions do not look like ordinary Type Ia supernovae, or any other
frequently observed transient.
