In this work, we discuss the neutrino masses and mixings as the realization of an S3 flavour permutational symmetry in two models, namely the Standard Model and an extension of the Standard Model with three Higgs doublets. In the S3 Standard Model, mass matrices of the same generic form are obtained for the neutrino and charged leptons when the S3 flavour symmetry is broken sequentially according to the chain S3L ⊗ S3R ⊃ S3diag ⊃ S2. In the minimal S3-symmetric extension of the Standard Model, the S3 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 neutrinos 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 S3 Standard Model, from a χ2 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 S3 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 θ23 in excellent agreement with experimental data, and θ13 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 θ13also 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.