While the core collapse supernova mechanism is uncertain, overwhelming evidence suggests that it is inherently multidimensional. This, plus the complexity of the physical processes involved and the increasing evidence from computation that the explosion may be marginal, presents great computational challenges for realistic modeling, particularly in 3 spatial dimensions. We have developed a code which couples PPM Lagrangian with remap hydrodynamics (Colella, P., & Woodward, P. R. 1984, J. Comput. Phys., 54, 174) , multigroup, flux-limited diffusion neutrino transport (Bruenn, S. W. 1985, ApJS, 58, 771, with many improvements), and a nuclear network (Hix, W. R., & Thielemann F.-K. 1996, ApJ, 460, 869.) The neutrino transport is performed in a ray-by-ray plus approximation wherein all the lateral effects of neutrinos are included (e.g., pressure, velocity corrections, advection) except the transport. A moving radial grid option permits the evolution to be carried out from initial core collapse with only modest demands on the number of radial zones. The inner part of the core is evolved after collapse along with the rest of the core and mantle by subcycling the lateral evolution near the center as demanded by the small Courant times. We present results of 2-D simulations of a symmetric and an asymmetric collapse of both a 15 and an 11 solar mass progenitor, and preliminary 3-D simulations of an 11 solar mass progenitor.