  Models of core-collapse supernova explosions powered by the neutrino-driven
mechanism have matured considerable in recent years. Explosions at the low-mass
end of the progenitor spectrum can routinely be simulated in 1D, 2D, and 3D and
allow us to study supernova nucleosynthesis based on first-principle models.
Results of nucleosynthesis calculations indicate that supernovae of the lowest
masses could be important contributors of some lighter n-rich elements beyond
iron. The explosion mechanism of more massive stars is still under
investigation, although first 3D models of neutrino-driven explosions employing
multi-group neutrino transport have recently become available. Together with
earlier 2D models and more simplified 3D simulations, these have elucidated the
interplay between neutrino heating and hydrodynamic instabilities in the
post-shock region that is essential for shock revival. However, some physical
ingredients may still need to be added or improved before simulations can
robustly explain supernova explosions over a wide mass range. We explore
possible issues that may affect the accuracy of supernova simulations, and
review some of the ideas that have recently been explored as avenues to robust
explosions, including uncertainties in the neutrino rates, rapid rotation, and
an external forcing of non-radial fluid motions by strong seed perturbations
from convective shell burning. The perturbation-aided neutrino-driven mechanism
and the implications of recent 3D simulations of shell burning in supernova
progenitors are discussed in detail. The efficacy of the perturbation-aided
mechanism is illustrated by the first successful multi-group neutrino
hydrodynamics simulation of an 18 solar mass progenitor with 3D initial
conditions. We conclude with speculations about the potential impact of 3D
effects on the structure of massive stars through convective boundary mixing.
(abridged)
