Type Ia supernovae (SNe Ia) have provided the first evidence that the
Universe is accelerating. Nevertheless, the nature of the progenitors
has remained a mystery, indeed controversial. In this talk I will
first summarize the main supernova explosion mechanisms and the
observational classification of supernovae. I will show how SNe Ia can
be used as standardizable distance candles and discuss possible
limitations of the method. In the main part of the talk I will
discuss the main progenitor models that have been proposed,
emphasizing the problems and advantages of each, and how they can be
observationally tested. I will then present some relatively recent
discoveries by the Palomar Transient Factory that directly constrain
the progenitors in a few cases.