The Symmetry TFT of a quantum field theory is a Topological Field Theory in one dimension more, that describes the structure of the symmetry, its representations, its anomalies, and the possible topological manipulations (gaugings). Until recently only the case of finite symmetries had been addressed. I will present a proposal for the Symmetry TFT of theories with a U(1) symmetry: it is a BF theory with a continuum of topological operators. I will discuss many examples, including chiral anomalies, 2-groups, the non-invertible Q/Z chiral symmetry in 4d with ABJ anomaly, and I will discuss the non-Abelian case as well.