Type Ia supernovae are important, but mysterious cosmological tools. Their standard brightnesses have enabled cosmologists to measure extreme distances and to discover dark energy. However, the nature of their progenitor mechanisms remains elusive, with many competing models offering only partial clues to their origins. Here, type Ia supernova delay times are explored using analytical models. Combined with a new observation technique, this model places new constraints on the characteristic time delay between the formation of stars and the first type Ia supernovae. This derived delay time (500 million years) implies low-mass companions for single degenerate progenitor scenarios. In the latter portions of this dissertation, two progenitor mechanisms are simulated in detail; white dwarf collisions and mergers. From the first of these simulations, it is evident that white dwarf collisions offer a viable and unique pathway to producing type Ia supernovae. Many of the combinations of masses simulated produce sufficient quantities of 56Ni (up to 0.51 solar masses) to masquerade as normal type Ia supernovae. Other combinations of masses produce 56Ni yields that span the entire range of supernova brightnesses, from the very dim and underluminous, with 0.14 solar masses, to the over-bright and superluminous, with up to 1.71 solar masses. The 56Ni yield in the collision simulations depends non-linearly on total system mass, mass ratio, and impact parameter. Using the same numerical tools as in the collisions examination, white dwarf mergers are studied in detail. Nearly all of the simulations produce merger remnants consisting of a cold, degenerate core surrounded by a hot accretion disk. The properties of these disks have strong implications for various viscosity treatments that have attempted to pin down the accretion times. Some mass combinations produce super-Chandrasekhar cores on shorter time scales than viscosity driven accretion. A handful of simulations also exhibit helium detonations on the surface of the primary that bear a resemblance to helium novae. Finally, some of the preliminary groundwork that has been laid for constructing a new numerical tool is discussed. This new tool advances the merger simulations further than any research group has done before, and has the potential to answer some of the lingering questions that the merger study has uncovered. The results of thermal diffusion tests using this tool have a remarkable correspondence to analytical predictions.