At the confluence of much of 20th-Century physics, lies an astrophysical puzzle that has taxed theorists and the computational arts for almost half a century.

Supernova explosions, the source of much of the heavy elements in the Universe and the birthplace of neutron stars and stellar-mass black holes, are still not understood. However, using sophisticated numerical tools and platforms, theorists have been able to conduct multi-dimensional simulations with increasing physical fidelity that have provided insight into the variety of phenonoma that attend stellar death and explosion. The core of the emerging theoretical synthesis is the centrality of asphericity and the breaking of spherical symmetry. In this talk, I will review the state of the field, the contending explosion models, and the connections with other exotic objects, such as gamma-ray bursts and hypernovae. In the process, I will highlight the state-of-the-art computational astrophysics which has been applied to date, and which may be necessary in the future, to credibly unravel this mystery.

"A New Mechanism for Core-Collapse Supernova Explosions," (A. Burrows, E. Livne, L. Dessart, C.D. Ott, & J. Murphy), Ap.J., 640, 878, 2006 (astro-ph/0510687).

"Features of the Acoustic Mechanism of Core-Collapse Supernova Explosions," (A. Burrows, E. Livne, L. Dessart, C.D. Ott, & J. Murphy), Ap.J. 655, 416, 2007 (astro-ph/0610175).

"A New Algorithm for 2-D Transport for Astrophysical Simulations: I. General Formulation and Tests for the 1-D Spherical Case," (I. Hubeny and A. Burrows), Ap.J., 659, 1458, 2007 (astro-ph/0609049).

"A Two-Dimensional Magnetohydrodynamics Scheme for General Unstructured Grids," (E. Livne, L. Dessart, A. Burrows, & C.A. Meakin), Ap.J. Suppl., 170, 187, 2007 (astro-ph/0702465).

"Simulations of Magnetically-Driven Supernova and Hypernova Explosions in the Context of Rapid Rotation," (A. Burrows, L. Dessart, E. Livne, C.D. Ott, & J. Murphy), Ap.J., 664, 416, 2007 (astro-ph/0702539).

"BETHE-Hydro: An Arbitrary Lagrangian-Eulerian Multi-dimensional Hydrodynamics Code for Astrophysical Simulations," (Jeremiah W. Murphy and Adam Burrows), Ap.J. Suppl., 179, 209, 2008 (arXiv:0805.3356).

"2D Multi-Angle, Multi-Group Neutrino Radiation-Hydrodynamic Simulations of Postbounce Supernova Cores," (C.D. Ott, A. Burrows, L. Dessart, & E. Livne), Ap.J., 685, 1069, 2008, (arxiv:0804.0239)