  Unraveling the mechanism for core-collapse supernova explosions is an
outstanding computational challenge and the problem remains essentially
unsolved despite more than four decades of effort. However, much progress in
realistic modeling has occurred recently through the availability of
multi-teraflop machines and the increasing sophistication of supernova codes.
These improvements have led to some key insights which may clarify the picture
in the not too distant future. Here we briefly review the current status of the
three explosion mechanisms (acoustic, MHD, and neutrino heating) that are
currently under active investigation, concentrating on the neutrino heating
mechanism as the one most likely responsible for producing explosions from
progenitors in the mass range ~10 to ~25 solar masses. We then briefly describe
the CHIMERA code, a supernova code we have developed to simulate core-collapse
supernovae in 1, 2, and 3 spatial dimensions. We finally describe the results
of an ongoing suite of 2D simulations initiated from a 12, 15, 20, and 25 solar
mass progenitor. These have all exhibited explosions and are currently in the
expanding phase with the shock at between 5,000 and 10,000 km. We finally very
briefly describe an ongoing simulation in 3 spatial dimensions initiated from
the 15 solar mass progenitor.
