We discuss the current status of progress towards unraveling the mystery of the corecollapse supernova mechanism. We advance the notion that the fundamental impediment to progress in supernova theory over several decades has been the lack of access to codes and computers with which to properly simidate the collapse phenomenon in 3D. There may be qualitative differences between ID, 2D, and 3D models. Since 3D capabiUties have only recently become a viable hope, this could explain theory's slow march since the 1960's towards demonstrating a robust mechanism of explosion. It has been shown that one-dimensional, spherical explosions reqiure a set of rare progenitor characteristics. Moreover, when they explode they are genericaUy underenergetic. Two-dimensional simidations are still ambiguous and problematic, though they manifest some of the midti-dimensional effects shown to enable explosion (Murphy & Burrows 2008). In this contribution, we summarize the recent residts from the various leading groups still engaged in exploring the explosion mechanism and describe our new 3D AMR radiation/hydrodynamics code CASTRO that promises to help address the remaining issues that surround this most central, if problematic, puzzle in theoretical astrophysics.