  Advances in our understanding and the modeling of stellar core-collapse and
supernova explosions over the past 15 years are reviewed, concentrating on the
evolution of hydrodynamical simulations, the description of weak interactions
and nuclear equation of state effects, and new insights into the
nucleosynthesis occurring in the early phases of the explosion, in particular
the neutrino-p process. The latter is enabled by the proton-richness of the
early ejecta, which was discovered because of significant progress has been
made in the treatment of neutrino transport and weak interactions. This
progress has led to a new generation of sophisticated Newtonian and
relativistic hydrodynamics simulations in spherical symmetry. Based on these,
it is now clear that the prompt bounce-shock mechanism is not the driver of
supernova explosions, and that the delayed neutrino-heating mechanism can
produce explosions without the aid of multi-dimensional processes only if the
progenitor star has an ONeMg core inside a very dilute He-core, i.e., has a
mass in the 8--10 solar mass range. Hydrodynamic instabilities of various kinds
have indeed been recognized to occur in the supernova core and to be of
potential importance for the explosion. Neutrino-driven explosions, however,
have been seen in two-dimensional simulations with sophisticated neutrino
transport so far only when the star has a small iron core and low density in
the surrounding shells as being found in stars near 10--11 solar masses. The
explosion mechanism of more massive progenitors is still a puzzle. It might
involve effects of three-dimensional hydrodynamics or might point to the
relevance of rapid rotation and magnetohydrodynamics, or to still incompletely
explored properties of neutrinos and the high-density equation of state.
