  One of the challenges to increasing the mass of a white dwarf through
accretion is the tendency for the accumulating hydrogen to ignite unstably and
potentially trigger mass loss. It has been known for many years that there is a
narrow range of accretion rates for which the hydrogen can burn stably,
allowing for the white dwarf mass to increase as a pure helium layer
accumulates. We first review the physics of stable burning, providing a clear
explanation for why radiation pressure stabilization leads to a narrow range of
accretion rates for stable burning near the Eddington limit, confirming the
recent work of Nomoto and collaborators. We also explore the possibility of
stabilization due to a high luminosity from beneath the burning layer. We then
examine the impact of the beta-decay-limited ''hot'' CNO cycle on the stability
of burning. Though this plays a significant role for accreting neutron stars,
we find that for accreting white dwarfs, it can only increase the range of
stably-burning accretion rates for metallicities below 0.01 solar metallicity.
