  Progress in the theory of stellar convection over the past decade is
reviewed. The similarities and differences between convection in stellar
envelopes and laboratory convection at high Rayleigh numbers are discussed.
Direct numerical simulation of the solar surface layers, with no other input
than atomic physics, the equations of hydrodynamics and radiative transfer is
now capable of reproducing the observed heat flux, convection velocities,
granulation patterns and line profiles with remarkably accuracy. These results
show that convection in stellar envelopes is an essentially non-local process,
being driven by cooling at the surface. This differs distinctly from the
traditional view of stellar convection in terms of local concepts such as
cascades of eddies in a mean superadiabatic gradient. The consequences this has
for our physical picture of processes in the convective envelope are
illustrated with the problems of sunspot heat flux blocking, the eruption of
magnetic flux from the base of the convection zone, and the Lithium depletion
problem.
