A principal `supernova neutrino challenge' is the computational difficulty of six-dimensional neutrino radiation hydrodynamics. The variety of resulting approximations has provoked a long history of uncertainty in the core-collapse supernova explosion mechanism, but recent work highlighting low-mode convection and a newly-recognized instability in spherical accretion shocks may signal (yet another) resolution. As part of its goal of elucidating the explosion mechanism, the Terascale Supernova Initiative is committed to meeting the full complexity of the computational challenge. The understanding of supernova neutrino emission gained in detailed simulations provides a potential basis for learning about two major remaining unknowns in neutrino flavor mixing: the value of the mixing angle $��_{13}$, and distinguishing between ``normal'' and ``inverted'' mass hierarchies.

6 pages. Contribution to the proceedings of NOW2004, Conca Specchiulla (Otranto, Italy), September 11-17, 2004, to be published by Nucl. Phys. B (Proc. Suppl.), ed. P. Bernardini, G.L. Fogli, and E. Lisi