  This is a status report on our endeavor to reveal the mechanism of
core-collapse supernovae (CCSNe) by large-scale numerical simulations.
Multi-dimensionality of the supernova engine, general relativistic
magnetohydrodynamics, energy and lepton number transport by neutrinos emitted
from the forming neutron star as well as nuclear interactions there, are all
believed to play crucial roles in repelling infalling matter and producing
energetic explosions. These ingredients are nonlinearly coupled with one
another in the dynamics of core-collapse, bounce, and shock expansion. Serious
quantitative studies of CCSNe hence make extensive numerical computations
mandatory. Since neutrinos are neither in thermal nor in chemical equilibrium
in general, their distributions in the phase space should be computed. This is
a six dimensional (6D) neutrino transport problem and quite a challenge even
for those with an access to the most advanced numerical resources such as the
"K computer". To tackle this problem, we have embarked on multi-front efforts.
In particular we report in this paper our recent progresses in the treatments
of multi-dimensional (multi-D) radiation-hydrodynamics. We are currently
proceeding on two different paths to the ultimate goal; in one approach we
employ an approximate but highly efficient scheme for neutrino transport and
treat 3D hydrodynamics and/or general relativity rigorously; some
neutrino-driven explosions will be presented and comparisons will be made
between 2D and 3D models quantitatively; in the second approach, on the other
hand, exact but so far Newtonian Boltzmann equations are solved in two and
three spatial dimensions; we will show some demonstrative test simulations. We
will also address the perspectives of exa-scale computations on the next
generation supercomputers.

High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;Nuclear Theory