  A quantitative understanding of the weak nuclear response is a prerequisite
for the computer simulations of astrophysical phenomena like supernov$\ae$
explosions and neutron star cooling. In order to reduce the systematic
uncertainties associated with the simulations, a consistent framework, able to
take into account dynamical correlation effects, is needed to compute
neutrino-nucleon and neutrino-nucleus reaction rates. In this paper we describe
the many-body theory of the weak nuclear response at low energy regime. We show
how to include both short and long correlations effects in a consistent
fashion.
