Type Ia supernovae (SNe Ia) play an important role in astrophysics and are crucial for the studies of stellar evolution, galaxy evolution and cosmology. They are generally thought to be thermonuclear explosions of accreting carbon-oxygen white dwarfs (CO WDs) in close binaries, however, the nature of the mass donor star is still unclear. In this article, we review various progenitor models proposed in the past years and summarize many observational results that can be used to put constraints on the nature of their progenitors. We also discuss the origin of SN Ia diversity and the impacts of SN Ia progenitors on some fields. The currently favourable progenitor model is the single-degenerate (SD) model, in which the WD accretes material from a non-degenerate companion star. This model may explain the similarities of most SNe Ia. It has long been argued that the double-degenerate (DD) model, which involves the merger of two CO WDs, may lead to an accretion-induced collapse rather than a thermonuclear explosion. However, recent observations of a few SNe Ia seem to support the DD model, and this model can produce normal SN Ia explosion under certain conditions. Additionally, the sub-luminous SNe Ia may be explained by the sub-Chandrasekhar mass model. At present, it seems likely that more than one progenitor model, including some variants of the SD and DD models, may be required to explain the observed diversity of SNe Ia.