Abstract
Most supernova explosions accompany the death of a massive star. These explosions give birth to neutron stars and black holes, and eject solar masses of heavy elements. However, determining the mechanism of explosion has been a half-century journey of great numerical and physical complexity. Here we present the status of this theoretical quest and the physics and astrophysics upon which its resolution seems to depend. The delayed neutrino-heating mechanism is emerging as the key driver of supernova explosions, but there remain many issues to address, such as the chaos of the involved dynamics.