  The flavour puzzle is an open problem both in the Standard Model and in its
possible supersymmetric or grand unified extensions. In this thesis, we discuss
possible explanations of the origin of fermion mass hierarchies and mixings by
the use of non-Abelian discrete flavour symmetries. We present a T'- and an
S4-based realisations in which the spontaneous breaking of the symmetry
produces realistic fermion mass hierarchies, the quark mixing matrix comparable
to the Wolfenstein parametrisation and the lepton mixing matrix close to the
so-called tribimaximal pattern. We also present an alternative S4-based flavour
model which naturally leads to the weak complementarity relation with a
relatively large lepton reactor angle. Masses and mixings are evaluated at a
very high energy scale and for a comparison with experimental measurements we
illustrate a general analysis on the stability under the renormalisation group
running to evolve these observables to low energies. We consider also the
constraints on flavour violating processes arising from introducing a flavour
symmetry: in particular we concentrate on the lepton sector, analysing some
lepton flavour violating decays and the discrepancy between the theoretical
prediction and the experimental measurement of the anomalous magnetic moment of
the muon. Interesting hints for the scale of New Physics and for the
forthcoming experimental results from LHC are found. Finally we discuss the
impact of an underlining flavour symmetry on leptogenesis in order to explain
the baryon asymmetry of the universe.
