  The current picture of the collapse and explosion of massive stars and the
formation of neutron stars is reviewed. According to the favored scenario,
however by no means proven and undisputed, neutrinos deposit the energy of the
explosion in the stellar medium which surrounds the nascent neutron star.
Observations, in particular of Supernova~1987A, suggest that mixing processes
play an important role in the expanding star, and multi-dimensional simulations
show that these are linked to convective instabilities in the immediate
vicinity of the neutron star. Convectively enhanced energy transport inside the
neutron star can have important consequences for the neutrino emission and thus
the neutrino-heating mechanism. This also holds for a suppression of the
neutrino interactions at nuclear densities. Multi-dimensional hydrodynamics,
general relativity, and a better understanding of the neutrino interactions in
neutron star matter may be crucial to resolve the problem that state-of-the-art
spherical models do not yield explosions even with a very accurate treatment of
neutrino transport by solving the Boltzmann equation.
