  We present theoretical results on the non flux-averaged
$^{208}Pb(\nu_{e},e^-)^{208}Bi$ and $^{208}Pb(\nu_{\mu},\mu^-)^{208}Bi$
reaction cross sections, obtained within the charge-exchange
Random-Phase-Approximation. A detailed knowledge of these cross sections is
important in different contexts. In particular, it is necessary to assess the
possibility of using lead as a detector in future experiments on supernova
neutrinos, such as OMNIS and LAND, and eventually detect neutrino oscillation
signals by exploiting the spectroscopic properties of $^{208}Bi$. We discuss
the present status on the theoretical predictions of the reaction cross
sections.
