Collapse-driven supernovae: When do they explode?

I will give an overview of the current status of our understanding on the machanism of collapse-driven supernovae and related phenomena, presenting some of our group's recent results. Included among the issues discussed in this article are (1) 1D simulations with the most detailed neutrino transport, (2) the global asymmetry of supernovae and its implications for the explosion mechanism, (3) hydrodynamical instabilities after the shock stagnation, and (4) gravitational collapse of more massive stars.