  We report here on recent progress in understanding the birth conditions of
neutron stars and the way how supernovae explode. More sophisticated numerical
models have led to the discovery of new phenomena in the supernova core, for
example a generic hydrodynamic instability of the stagnant supernova shock
against low-mode nonradial deformation and the excitation of gravity-wave
activity in the surface and core of the nascent neutron star. Both can have
supportive or decisive influence on the inauguration of the explosion, the
former by improving the conditions for energy deposition by neutrino heating in
the postshock gas, the latter by supplying the developing blast with a flux of
acoustic power that adds to the energy transfer by neutrinos. While recent
two-dimensional models suggest that the neutrino-driven mechanism may be viable
for stars from about 8 solar masses to at least 15 solar masses, acoustic
energy input has been advocated as an alternative if neutrino heating fails.
Magnetohydrodynamic effects constitute another way to trigger explosions in
connection with the collapse of sufficiently rapidly rotating stellar cores,
perhaps linked to the birth of magnetars. The global explosion asymmetries seen
in the recent simulations offer an explanation of even the highest measured
kick velocities of young neutron stars.
