Many stellar binaries are orbiting so closely that matter flows from one object to another. When the accreting star is a white dwarf, the accumulated hydrogen eventually ignites nuclear reactions that trigger a thermonuclear runaway called a classical novae. These events can be seen in nearby galaxies and provide our best inventory of the number of such binaries. For some special cases, the accumulating hydrogen can stably burn, allowing for the white dwarf to increase in mass and central density to the point of a central ignition of carbon fusion. The resulting disruptive explosion is the cause of Type Ia supernovae, events so bright that we can detect them across the universe. I will discuss new results on the study of these supernovae and their occurrence in different types of galaxies, both young and old.