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Core collapse supernovae are thought to be powered not dynamically by the direct piston mechanism of core bounce, but by neutrino heating after the bounce shock has stalled into accretion. The theory we will describe provides a paradigm for understanding the neutrino heating mechanism. Both the critical condition for instability and the explosion energy are very steep functions of the driving neutrino luminosity. We will present recent 1D and 2D hydrodynamic calculations in which the basics of the supernova mechanism are elucidated. The shock wave that stalls within ten milliseconds of its creation during the collapse and bounce of the core of a massive star leaves behind it unstable lepton and entropy profiles that can drive a violent Rayleigh-Taylor overturn. Furthermore, the core neutrino luminosities can establish unstable entropy gradients near the shock 50 milliseconds after bounce. John Hayes, Bruce Fryxell and I have demonstrated the possible existence of a convective boost in the neutrino luminosities due to core lepton overturn that can ignite a supernova explosion and have verified the potential importance of $\nu$-driven ``Bethe'' convection near the shock. Issues that surround the residual neutron star mass, the $^{56}$Ni yields, the supernova energies, the progenitors, and the nucleosynthetic consequences will also be addressed.
