Core-collapse supernovae are a puzzle that has taxed
theorists and computational science for half a century.
Such explosions are the source of many of the heavy elements
in the Universe and the birthplace of neutron stars and
stellar-mass black holes. However, determining the mechanism
of explosion remains the key goal of theory. Recently,
using sophisticated numerical tools and platforms, theorists
have been able to conduct multi-dimensional simulations
with some physical fidelity that have provided insight
into the phenonoma that attend stellar death
and explosion.  The core of the emerging theoretical
synthesis is the centrality of spatial dimension in the context
of hydrodynamic instability and asphericity.  In this talk, I will
review the state of the field and the contending explosion models. In
the process, I will highlight the computational astrophysics
that has been applied to date, and that may be necessary
in the future to credibly unravel this mystery.