In spite of several decades of progress in the development of numerical hydrodynamical simulations, the modelling of stellar convection is still considered to be an unresolved question in astrophysics. This is also because numerical simulations can account for only a limited range of scales in space and time and like in climatology, a more flexible approach is needed which integrates different models applied on different scales. Overshooting and its effects on mixing in stellar interiors is a particular example for this general problem in convection modelling. In the presentation, some recent advances on the modelling side and on the numerical simulation side are discussed, for the case of overshooting in convective stellar envelopes and convective stellar cores. Precision asteroseismology is argued to be essential to decide on the physical completeness and applicability of a convection model, particularly for cases where the simulations themselves have to be considered physically incomplete.

List of Collaborators given on the introductory slides. Contributions of figures acknowledged on each slide.