Application of non-Abelian finite groups to the theory of neutrino masses and mixing is reviewed, which is strongly suggested by the agreement of the tribimaximal ͑TB͒ mixing pattern with experiment. After summarizing the motivation and the formalism, concrete models based on A 4 , S 4 , and other finite groups, and their phenomenological implications are discussed, including lepton flavor violating processes, leptogenesis, and the extension to quarks. As an alternative to TB mixing application of discrete flavor symmetries to quark-lepton complementarity and bimaximal mixing is also considered.

We formulate and discuss a 4-dimensional SUSY version of an A 4 model for tribimaximal neutrino mixing which is completely natural. We also study the next-tothe-leading corrections and show that they are small, once the ratios of A 4 breaking VEVs to the cutoff are fixed in a specified interval. We also point out an interesting way of presenting the A 4 group starting from the modular group. In this approach, which could be interesting in itself as an indication on a possible origin of A 4 , the lagrangian basis where the symmetry is formulated coincides with the basis where the charged leptons are diagonal. If the same classification structure in A 4 is extended from leptons to quarks, the CKM matrix coincides with the unit matrix in leading order and a study of non leading corrections shows that the departures from unity of the CKM matrix are far too small to accomodate the observed mixing angles.

The approximate symmetries of Quantum ChromoDynamics in the infinite heavy quark (Q = c, b) mass limit (mQ → ∞) and in the chiral limit for the light quarks (mq → 0, q = u, d, s) can be used together to build up an effective chiral lagrangian for heavy and light mesons describing strong interactions among effective meson fields as well as their couplings to electromagnetic and weak currents, including the relevant symmetry breaking terms. The effective theory includes heavy (Qq) mesons of both negative and positive parity, light pseudoscalars, as well as light vector mesons. We summarize the estimates for the parameters entering the effective lagrangian and discuss in particular some phenomenologically important couplings, such as gB * Bπ. The hyperfine splitting of heavy mesons is discussed in detail. The effective lagrangian allows for the possibility to describe consistently weak couplings of heavy (B, D) to light (π, ρ, K * , etc.) mesons. The method has however its own limitations, due to the requirement that the light meson momenta should be small, and we discuss how such limitations can be circumvented through reasonable ansatz on the form factors. Flavour conserving (e. g. B * → B γ) and flavour changing (e. g. B → K * γ) radiative decays provide another field of applications of effective lagrangians; they are discussed together with their phenomenological implications. Finally we analyze effective lagrangians describing heavy charmonium-like (QQ) mesons and their strong and electromagnetic interactions. The role of approximate heavy quark symmetries for this case and the phenomenological tests of these models are also discussed.