  We briefly summarize recent efforts in Garching for modeling stellar core
collapse and post-bounce evolution in one and two dimensions. The transport of
neutrinos of all flavors is treated by iteratively solving the coupled system
of frequency-dependent moment equations together with a model Boltzmann
equation which provides the closure. A variety of progenitor stars, different
nuclear equations of state, stellar rotation, and global asymmetries due to
large-mode hydrodynamic instabilities have been investigated to ascertain the
road to finally successful, convectively supported neutrino-driven explosions.
