  The results of recent multi-dimensional simulations of type-II supernovae are
reviewed. They show that convective instabilities in the collapsed stellar core
might play an important role already during the first second after the
formation of the supernova shock. Convectively unstable situations occur below
and near the neutrinosphere as well as in the neutrino-heated region between
the nascent neutron star and the supernova shock after the latter has stalled
at a radius of typically 100-200 km.
  While convective overturn in the layer of neutrino energy deposition clearly
helps the explosion to develop and potentially provides an explanation of
strong mantle and envelope mixing, asphericities, and non-uniform 56Ni
distribution observed in supernova SN 1987A, its presence and importance
depends on the strength of the neutrino heating and thus on the size of the
neutrino fluxes from the neutron star. Convection in the hot-bubble region can
only develop if the growth timescale of the instabilities and the heating
timescale are both shorter than the accretion timescale of the matter advected
through the stagnant shock. For too small neutrino luminosities this
requirement is not fulfilled and convective activity cannot develop, leading to
very weak explosions or even fizzling models, just as in the one-dimensional
situation.
  Convectively enhanced neutrino luminosities from the protoneutron star can
therefore provide an essential condition for the explosion of the star. Very
recent two-dimensional, self-consistent, general relativistic simulations of
the cooling of a newly-formed neutron star demonstrate and confirm the
possibility that Ledoux convection, driven by negative lepton number and
