  Core-collapse supernovae are among the most fascinating phenomena in
astrophysics and provide a formidable challenge for theoretical investigation.
They mark the spectacular end of the lives of massive stars and, in an
explosive eruption, release as much energy as the sun produces during its whole
life. A better understanding of the astrophysical role of supernovae as birth
sites of neutron stars, black holes, and heavy chemical elements, and more
reliable predictions of the observable signals from stellar death events are
tightly linked to the solution of the long-standing puzzle how collapsing stars
achieve to explode. In this article our current knowledge of the processes that
contribute to the success of the explosion mechanism are concisely reviewed.
After a short overview of the sequence of stages of stellar core-collapse
events, the general properties of the progenitor-dependent neutrino emission
will be briefly described. Applying sophisticated neutrino transport in
axisymmetric (2D) simulations with general relativity as well as in simulations
with an approximate treatment of relativistic effects, we could find successful
neutrino-driven explosions for a growing set of progenitor stars. First results
of three-dimensional (3D) models have been obtained, and magnetohydrodynamic
simulations demonstrate that strong initial magnetic fields in the pre-collapse
core can foster the onset of neutrino-powered supernova explosions even in
nonrotating stars. These results are discussed in the context of the present
controversy about the value of 2D simulations for exploring the supernova
mechanism in realistic 3D environments, and they are interpreted against the
background of the current disagreement on the question whether the standing
accretion shock instability (SASI) or neutrino-driven convection is the crucial
agency that supports the onset of the explosion.
