Neutrino mass and mixing with modular symmetry.

This is a review article about neutrino mass and mixing and flavour model building strategies based on modular symmetry. After a brief survey of neutrino mass and lepton mixing, and various Majorana seesaw mechanisms, we construct and parameterise the lepton mixing matrix and summarise the latest global fits, before discussing the flavour problem of the Standard Model. We then introduce some simple patterns of lepton mixing, introduce family (or flavour) symmetries, and show how they may be applied to direct, semi-direct and tri-direct CP models, where the simple patterns of lepton mixing, or corrected versions of them, may be enforced by the full family symmetry or a part of it, leading to mixing sum rules. We then turn to the main subject of this review, namely a pedagogical introduction to modular symmetry as a candidate for family symmetry, from the bottom-up point of view. After an informal introduction to modular symmetry, we introduce the modular group, and discuss its fixed points and residual symmetry, assuming supersymmetry throughout. We then introduce finite modular groups of leveland modular forms with integer or rational modular weights, corresponding to simple geometric groups or their double or metaplectic covers, including the most general finite modular groups and vector-valued modular forms, with detailed results forN=2,3,4,5. The interplay between modular symmetry and generalized CP symmetry is discussed, deriving CP transformations on matter multiplets and modular forms, highlighting the CP fixed points and their implications. In general, compactification of extra dimensions generally leads to a number of moduli, and modular invariance with factorizable and non-factorizable multiple moduli based on symplectic modular invariance and automorphic forms is reviewed. Modular strategies for understanding fermion mass hierarchies are discussed, including the weighton mechanism, small deviations from fixed points, and texture zeroes. Then examples of modular models are discussed based on single modulusmodels, a minimalS4'model of leptons (and quarks), and a multiple moduli model based on threegroups capable of reproducing the Littlest Seesaw model. We then extend the discussion to include Grand Unified Theories based on modular (flipped)(5) and(10). Finally we briefly mention some issues related to top-down approaches based on string theory, including eclectic flavour symmetry and moduli stabilisation, before concluding.