When a massive star reaches the end of its life, the core of the star collapses to a neutron star or black hole while the outer stellar layers are expelled in a supernova explosion. These cosmic catastrophies are not only among the most spectacular celestial phenomena, they are also responsible for the production and dissemination of a major part of the heavy elements in the universe. A better understanding of the role of supernovae in astrophysics and as laboratories for nuclear and particle physics at extreme conditions requires the solution of one of the most long-standing problems of stellar physics: What is the mechanism that initiates and powers the explosion of stars? Increasingly sophisticated numerical models provide growing support that the energy deposition by neutrinos radiated from the hot, newly formed neutron star and aided by violent hydrodynamic mass motions is the driving agency of the explosion. In this talk I will review recent successes of theoretical modeling and new questions arising as simulations currently push forward to meet the grand computational challenges of the third spatial dimension. I will also discuss possibilities to confront the theoretical picture with observational tests and constraints.