Friedrich Roepke - March 16, 2018 Simulations of Type Ia supernova explosions Type Ia supernovae are among the most prominent cosmic explosions in the current universe. Because of their extreme brightness and correlations between observable properties that allow for a calibration as distance indicators, they have been instrumental in surveying its geometry. At the same time, these explosions are prolific sources of iron group elements and shape the chemical structure of their host galaxies. Despite their significance for various astrophysical processes, the mechanism of Type Ia supernovae remains enigmatic. A fundamental handicap for model building is the unknown nature of the progenitor system. Several distinct scenarios are currently discussed involving explosions of Chandrasekhar-mass and sub-Chandrasekhar mass white dwarf stars. These can be tested in numerical simulations, provided that uncertainties in the physical explosion mechanism and numerical approximations can be controlled. This is a very challenging task because of the pronounced multi-scale multi-physics character of the explosion process. Nonetheless, significant progress was possible with multidimensional hydrodynamic explosion simulations, that provide the basis for the prediction of observables. I will review the involved challenges, describe possible approaches to model building and numerical implementation, and give examples for simulations of thermonuclear supernova explosions in different scenarios. Finally, I will discuss achievements and shortcomings of current multidimensional models and outline paths to validate them based on comparison with various observational data.