Type Ia supernovae (SNe Ia), resulting from the detonation of white dwarfs (WDs), play an important role as standardizable candles for cosmology. Yet, the nature of their stellar progenitors remains a largely unsolved problem. In this talk, I will present recent multidimensional simulations of both the single and double-degenerate channels of SNe Ia. I will also discuss the use of late-time (> 1000 d past maximum) light curves to constrain both the progenitor WD masses as well as the uncertain ignition and detonation mechanisms of SNe Ia. I will conclude with implications for the SN Ia stellar progenitor problem.