Type Ia supernovae (SNe Ia) play a key role in the fields of astrophysics and cosmology. It is widely accepted that SNe Ia arise from thermonuclear explosions of white dwarfs (WDs) in binary systems. However, there is no consensus on the fundamental aspects of the nature of SN Ia progenitors and their actual explosion mechanism. This fundamentally flaws our understanding of these important astrophysical objects. In this review, we outline the diversity of SNe Ia and the proposed progenitor models and explosion mechanisms. We discuss the recent theoretical and observational progress in addressing the SN Ia progenitor and explosion mechanism in terms of the observables at various stages of the explosion, including rates and delay times, pre-explosion companion stars, ejecta-companion interaction, early excess emission, early radio/X-ray emission from circumstellar material interaction, surviving companion stars, late-time spectra and photometry, polarization signals and supernova remnant properties. Despite the efforts from both the theoretical and observational sides, questions of how the WDs reach an explosive state and what progenitor systems are more likely to produce SNe Ia remain open. No single published model is able to consistently explain all observational features and the full diversity of SNe Ia. This may indicate that either a new progenitor paradigm or an improvement in current models is needed if all SNe Ia arise from the same origin. An alternative scenario is that different progenitor channels and explosion mechanisms contribute to SNe Ia. In the next decade, the ongoing campaigns with the James Webb Space Telescope, Gaia and the Zwicky Transient Facility, and upcoming extensive projects with the Vera C. Rubin Observatory's Legacy Survey of Space and Time and the Square Kilometre Array will allow us to conduct not only studies of individual SNe Ia in unprecedented detail but also systematic investigations for different subclasses of SNe Ia. This will advance theory and observations of SNe Ia sufficiently far to gain a deeper understanding of their origin and explosion mechanism.