Because calibrated light curves of type Ia supernovae have become a major tool to determine the local expansion rate of the Universe, considerable attention has been given to models of these events over the past couple of years. It is now common believe that perhaps most type Ia supernovae are the explosions of white dwarfs that have approached the Chandrasekhar mass, MChan, ~ 1.39 Msolar, and are disrupted by thermonuclear fusion of carbon and oxygen. However, the mechanism whereby such accreting carbon-oxygen white dwarfs explode continues to be uncertain. Recent progress in modeling type Ia supernovae as well as several of the still open questions are addressed in this article. Although the main emphasis will be on studies of the explosion mechanism itself and on the related physical processes, including the physics and nuclear physics of turbulent nuclear combustion in degenerate stars, observational implications and constraints will also be discussed.