  We discuss the neutrino masses and mixings as the realization of an $S_{3}$
flavour permutational symmetry in two models, namely the Standard Model and an
extension of the Standard Model with three Higgs doublets. In the $S_3$
Standard Model, mass matrices of the same generic form are obtained for the
neutrinos and charged leptons when the $S_{3}$ flavour symmetry is broken
sequentially. In the minimal $S_{3}$-symmetric extension of the Standard Model,
the $S_3$ symmetry is left unbroken, and the concept of flavour is extended to
the Higgs sector by introducing in the theory three Higgs fields which are
SU(2) doublets. In both models, the mass matrices of the neutrino and charged
leptons are reparametrized in terms of their eigenvalues, and exact, explicit
analytical expressions for the neutrino mixing angles as functions of the
masses of neutrinos and charged leptons are obtained. In the case of the $S_3$
Standard Model, from a $\chi^{2}$ fit of the theoretical expressions of the
lepton mixing matrix to the values extracted from experiment, the numerical
values of the neutrino mixing angles are obtained in excellent agreement with
experimental data. In the $S_3$ extension of the Standard Model, if two of the
right handed neutrinos masses are degenerate, the reactor and atmospheric
mixing angles are determined by the masses of the charged leptons, yielding
$\theta_{23}$ in excellent agreement with experimental data, and $\theta_{13}$
different from zero but very small. If the masses of the three right handed
neutrinos are assumed to be different, then it is possible to get $\theta_{13}$
also in very good agreement with experimental data. We also show the branching
ratios of some selected flavour changing neutral currents (FCNC) process as
well as the contribution of the exchange of a neutral flavour changing scalar
to the anomaly of the magnetic moment of the muon.
