  We extend the formalism of weak interaction processes, obtaining new
expressions for the transition rates, which greatly facilitate numerical
calculations, both for neutrino-nucleus reactions and muon capture. Explicit
violation of CVC hypothesis by the Coulomb field, as well as development of a
sum rule approach for the inclusive cross sections have been worked out. We
have done a thorough study of exclusive (ground state) properties of $^{12}$B
and $^{12}$N within the projected quasiparticle random phase approximation
(PQRPA). Good agreement with experimental data achieved in this way put in
evidence the limitations of standard RPA and the QRPA models, which come from
the inability of the RPA in opening the $p_{3/2}$ shell, and from the
non-conservation of the number of particles in the QRPA. The inclusive
neutrino/antineutrino ($\nu/\tilde{\nu}$) reactions $^{12}$C($\nu,e^-)^{12}$N
and $^{12}$C($\tilde{\nu},e^+)^{12}$B are calculated within both the PQRPA, and
the relativistic QRPA (RQRPA). It is found that the magnitudes of the resulting
cross-sections: i) are close to the sum-rule limit at low energy, but
significantly smaller than this limit at high energies both for $\nu$ and
$\tilde{\nu}$, ii) they steadily increase when the size of the configuration
space is augmented, and particulary for $\nu/\tilde{\nu}$ energies $> 200$ MeV,
and iii) converge for sufficiently large configuration space and final state
spin. The quasi-elastic $^{12}$C($\nu,\mu^-)^{12}$N cross section recently
measured in the MiniBooNE experiment is briefly discussed. We study the
decomposition of the inclusive cross-section based on the degree of
forbiddenness of different multipoles. A few words are dedicated to the
$\nu/\tilde{\nu}$-$^{12}$C charge-exchange reactions related with astrophysical
applications.
