In the standard supernova picture, Type Ib/c and Type II supernovae are powered by the potential energy released in the collapse of the core of a massive star. In studying supernovae, we primarily focus on the ejecta that makes it beyond the potential well of the collapsed core. But, as we shall show in this paper, inmost supernova explosions a tenth of a solar mass or more of the ejecta is decelerated enough that it does not escape the potential well of that compact object. This material falls back onto the proto-neutron star within the first 10-15 s after the launch of the explosion, releasing more than 10(52) erg of additional potential energy. Most of this energy is emitted in the form of neutrinos and we must understand this fallback neutrino emission if we are to use neutrino observations to study the behavior of matter at high densities. Here we present both a one-dimensional study of fallback using energy-injected, supernova explosions and a first study of neutrino emission from fallback using a suite of two-dimensional simulations.