  Multi-dimensional hydrodynamic simulations of the post-bounce evolution of
collapsed stellar iron cores have demonstrated that convective overturn between
the stalled shock and the neutrinosphere can have an important effect on the
neutrino-driven explosion mechanism. Whether a model yields a successful
explosion or not, however, still depends on the power of neutrino energy
deposition behind the stalled shock. The neutrino interaction with the stellar
gas in the ``hot bubble'' also determines the duration of the shock stagnation
phase, the explosion energy, and the composition of the neutrino-heated
supernova ejecta. More accurate models require a more precise calculation of
the neutrino luminosities and spectra and of the angular distributions of the
neutrinos in the heating region. Therefore it is necessary to improve the
numerical treatment of the neutrino transport, to develop a better
understanding of the neutrino opacities of the dense nuclear medium, and to
take into account convective processes {\it inside} the newly formed neutron
star.
