  High-resolution images of the solar surface show a granulation pattern of hot
rising and cooler downward-sinking material -- the top of the deep-reaching
solar convection zone. Convection plays a role for the thermal structure of the
solar interior and the dynamo acting there, for the stratification of the
photosphere, where most of the visible light is emitted, as well as for the
energy budget of the spectacular processes in the chromosphere and corona.
Convective stellar atmospheres can be modeled by numerically solving the
coupled equations of (magneto)hydrodynamics and non-local radiation transport
in the presence of a gravity field. The CO5BOLD code described in this article
is designed for so-called "realistic" simulations that take into account the
detailed microphysics under the conditions in solar or stellar surface layers
(equation-of-state and optical properties of the matter). These simulations
indeed deserve the label "realistic" because they reproduce the various
observables very well -- with only minor differences between different
implementations. The agreement with observations has improved over time and the
simulations are now well-established and have been performed for a number of
stars. Still, severe challenges are encountered when it comes to extending
these simulations to include ideally the entire star or substellar object: the
strong stratification leads to completely different conditions in the interior,
the photosphere, and the corona. Simulations have to account for a large range
of spatial and time scales and also non-equilibrium processes. Last but not
least, realistic simulations are based on detailed microphysics and depend on
the quality of the input data. This article provides an overview of the
physical problem and the numerical solution and the capabilities of CO5BOLD,
illustrated with a number of applications.
