Type Ia supernova explosions are thought to result from the runaway thermonuclear burning of a carbon-oxygen white dwarf on a timescale of approximately one second, and can outshine an entire galaxy of normal stars at peak brightness. It is of fundamental interest to understand the detailed mechanism for such gigantic explosions. But observations of distant Type Ia supernovae also provide the most direct evidence that the Universe is dominated on cosmological scales by a mysterious "dark energy", which produces an effective antigravity that accelerates the expansion of the Universe. Thus an improved understanding of the Type Ia supernova mechanism is of intrinsic fundamental interest, but also has large implications for observational cosmology. I will give an overview suitable for a non-specialist scientific audience of what we know about these remarkable events, and discuss our efforts to improve the large-scale computer simulation of Type Ia supernovae using 3D adaptive-mesh hydrodynamics and realistic thermonuclear burning networks.