Recent progress in modeling type Ia supernovae by means of 3-dimensional hydrodynamic simulations as well as several of the still open questions are addressed. Our models are based on the assumption that thermonuclear burning inside a Chandrasekhar-mass C+O white dwarf is similar to turbulent chemical combustion and that, thus, thermonuclear supernovae can be modeled by means of numerical methods which have been developed and tested for laboratory and technical flames. It is shown that the new models have considerable predictive power and allow to study observable properties of type Ia supernovae, such as their light curves and spectra, without adjustable non-physical parameters, and they make firm predictions for the nucleosynthesis yields from the explosions. This raises a quest for better data, covering the spectroscopical and photometric evolution in all wave bands from very early epochs all the way into the nebular phase. First such results obtained by the European Supernova Collaboration (ESC) for a sample of nearby SNe Ia and their implications for constraining the models and systematic differences between them are also discussed.