This is a review paper about neutrino mass and mixing and flavour model building strategies based on discrete family symmetry. After a pedagogical introduction and overview of the whole of neutrino physics, we focus on the PMNS mixing matrix and the latest global fits following the Daya Bay and RENO experiments which measure the reactor angle. We then describe the simple bimaximal, tri-bimaximal and golden ratio patterns of lepton mixing and the deviations required for a non-zero reactor angle, with solar or atmospheric mixing sum rules resulting from charged lepton corrections or residual trimaximal mixing. The different types of see-saw mechanism are then reviewed as well as the sequential dominance mechanism. We then give a mini-review of finite group theory, which may be used as a discrete family symmetry broken by flavons either completely, or with different subgroups preserved in the neutrino and charged lepton sectors. These two approaches are then reviewed in detail in separate chapters including mechanisms for flavon vacuum alignment and different model building strategies that have been proposed to generate the reactor angle. We then briefly review grand unified theories (GUTs) and how they may be combined with discrete family symmetry to describe all quark and lepton masses and mixing. Finally, we discuss three model examples which combine an SU(5) GUT with the discrete family symmetries A₄, S₄ and Δ(96).

During 2004, four divisions of the American Physical Society commissioned a study of neutrino physics to take stock of where the field is at the moment and where it is going in the near and far future. Several working groups looked at various aspects of this vast field. The summary was published as a main report entitled ``The Neutrino Matrix'' accompanied by short 50 page versions of the report of each working group. Theoretical research in this field has been quite extensive and touches many areas and the short 50 page report provided only a brief summary and overview of few of the important points. The theory discussion group felt that it may be of value to the community to publish the entire study as a white paper and the result is the current article. After a brief overview of the present knowledge of neutrino masses and mixing and some popular ways to probe the new physics implied by recent data, the white paper summarizes what can be learned about physics beyond the Standard Model from the various proposed neutrino experiments. It also comments on the impact of the experiments on our understanding of the origin of the matter-antimatter asymmetry of the Universe and the basic nature of neutrino interactions as well as the existence of possible additional neutrinos. Extensive references to original literature are provided.