Non-invertible symmetries are a recently discovered generalization of symmetry that goes beyond group actions and conserved currents, appearing in a wide range of quantum field theories. In this talk I will introduce these structures through concrete examples, mostly in 1+1 dimensions. I will first describe recent work in which we construct continuous non-invertible symmetries and prove a generalized Noether theorem which connects them to the existence of non-local conserved currents. I will then turn to quantum gravity, and resolve an apparent contradiction between the presence of non-invertible symmetries on the worldsheet of string theory and the longstanding “no global symmetries” conjecture. Throughout, I will emphasize physical applications and the broader implications for quantum field theory.