Type Ia supernovae (SNe Ia) correspond to the thermonuclear explosion of a carbon-oxygen white dwarf (C-O WD) star in a binary system, triggered by the accretion of material from another star, or the merger/collision with a secondary WD. Their phenomenal luminosity -- several billion times that of the sun -- has motivated their use as cosmological distance indicators and led to the discovery of the accelerated expansion of the universe. SNe Ia are also the main producers of iron and hence play a fundamental role in the chemical evolution of galaxies. While recent observations have confirmed the basic theoretical picture of an exploding C-O WD star whose luminosity is powered by the radioactive decay of $^{56}$Ni, a number of uncertainties remain concerning the nature of the binary companion and the explosion mechanism. Several lines of evidence point towards the existence of multiple progenitor channels in order to explain the full range of the observed diversity. A complete physical understanding of these energetic stellar explosions remains a long-lasting goal of modern astrophysics.