In this talk I show many aspects of gauged L_{mu-tau} symmetry. The symmetry gauges {the muon number – the tau number} in the standard model so as to be anomaly free. First I will show how it is interesting for phenomenology. It includes IceCube Gap, Hubble tension etc. Then I show our phenomenological model for lepton mass and mixing with the symmetry. In this model the symmetry appears as the family symmetry. Finally I will present my trial to unify the symmetry with the Standard gauge group in terms of coset space family unification.