In this thesis I present an updated and improved equation of state (which we call the NDL EoS) for use in neutron-star structure and supernova simulations. This EoS is based upon a framework originally developed by Bowers & Wilson, but there are numerous changes. Among them are:   a reformulation in the context of density functional theory;  the possibility of the formation of material with a net proton excess (Ye > 0.5);  an improved treatment of the nuclear statistical equilibrium and the transition to heavy nuclei as the density approaches nuclear matter density;  an improved treatment of the effects of pions in the regime above nuclear matter density;  the effects of 3-body nuclear forces at high densities; and  a first order or crossover transition to a QCD chiral symmetry restoration and deconfinement phase at densities above nuclear matter density.  This thesis details the physics of, and constraints on, this new EoS and describes its implementation in numerical simulations. I show comparisons of this EoS with other equations of state commonly used in supernova collapse and neutron star simulations. I also show the effect the NDL EoS has on the thermodynamic quantities and nuclear abundances in the collapse simulation of Mayle & Wilson.