Accretion onto compact stars (white dwarfs and neutron stars) can
lead to a wide variety of interesting phenomena, including X-ray bursts,
Novae, and Supernovae. These events all have in common a thermonuclear
flash resulting from the nuclear ignition of matter under degenerate conditions
and the formation and propagation of a burning front (either detonation
or deflagration). Understanding these events requires studying the
interaction of a large number of physical processes, including gas dynamics,
nuclear burning, radiation transfer, conduction, convection, turbulence,
gravity, and magnetohydrodynamics. This talk will present an overview
of work in progress to understand these events using numerical simulations.