Symmetries have long been a staple of theoretical physics. Recent developments have led to extensions of the notion of symmetry to so-called “generalized symmetries,” a particularly interesting class of which are "non-invertible" symmetries. Non-invertible symmetry is an intrinsically quantum form of symmetry, which has the interesting property that its symmetry transformations do not form a group. In this talk I will give an introduction to non-invertible symmetries, and discuss applications to both Condensed Matter (CM) and High Energy Physics (HEP) problems. Throughout the talk we will discover a deep relation between symmetry and topology, and will use this relation (in a separate guise) to make statements about string theory as well.