  We overview the progress of the tables of the equation of state for
astrophysical simulations and the numerical methods of neutrino transfer. Hot
and dense matter play essential roles in core-collapse supernovae and neutron
stars. Equation of state determines the structure of compact objects and their
dynamics through its behavior of thermodynamic quantities. In addition,
neutrinos are trapped in supernova cores and neutron star mergers and
frequently interact with matter to crucially affect dynamics in determining the
explosion mechanism and the final form of compact objects. Therefore, it is
essential to implement detailed processes of nuclear and neutrino physics in
numerical simulations by having reliable data set of the equation of state and
reaction rates. We show examples of developments of the equation of state and
the neutrino transfer and discuss research directions toward understanding the
explosive phenomena by the first principle calculation.
