Understanding Core-Collapse Supernovae

I summarize, in the form of an extended abstract, the ongoing efforts at the University of Arizona (and in collaboration) to understand core-collapse supernovae theoretically. Included are short discussions of 1D (SESAME) and 2D (VULCAN/2D) codes and results, as well as discussions of the possible role of rotation. Highlighted are recent developments in multi-dimensional radiation hydrodynamics and the essential physics of the neutrino-driven mechanism.