Many ongoing simulations of core-collapse supernova explosions utilize the Lattimer-Swesty equation of state (EOS) in which a four-component nuclear statistical equilibrium is employed. The components are neutrons, protons, alpha particles, and heavy nuclei. At densities less than nuclear saturation density, and at temperatures small enough that nuclei exist, the ensemble of heavy nuclei is approximated by the thermodynamically preferred nucleus -- the so-called single-nucleus approximation -- whose properties are estimated using a liquid-droplet model. At supra-nuclear densities, the EOS is determined by using Skyrme-like effective interactions. The phase boundary between phases containing nuclei and phases without nuclei are determined through Gibbs' rules for phase equilibrium. We have generalized the above approach to include (i) more general potential and effective field-theoretical models, (ii) ensembles of nuclei, (iii) effects of neutron-skins, and (ii) effects of pionic excitations in medium. Results of these developments along with methods of implementing our more general approach in supernova simulations will be presented.