We review some recent developments in theoretical studies on the connection between the progenitor systems of Type Ia supernovae (SNe Ia) and the explosion mechanisms. (1) DD-subCh: In the merging of double C+O white dwarfs (DD scenario), if the carbon detonation is induced near the white dwarf (WD) surface in the early dynamical phase, it could result in the (effectively) sub-Chandrasekhar mass explosion. (2) DD-Ch: If no surface C-detonation is ignited, the WD could grow until the Chandrasekhar mass is reached, but the outcome depends on whether the quiescent carbon shell burning is ignited and burns C+O into O+Ne+Mg. (3) SD-subCh: In the single degenerate (SD) scenario, if the He shell-flashes grow strong to induce a He detonation, it leads to the sub-Chandra explosion. (4) SD-Ch: If the He-shell flashes are not strong enough, they still produce interesting amounts of Si and S near the surface of the C+O WD before the explosion. In the Chandra mass explosion, the central density is high enough to produce electron capture elements, e.g., stable 58Ni. Observations of the emission lines of Ni in the nebular spectra provides useful diagnostics of the sub-Chandra vs. Chandra issue. The recent observations of relatively low velocity carbon near the surface of SNe Ia provide also an interesting constraint on the explosion models.