The paper explores the nature and implications of non-invertible symmetries in quantum field theory (QFT) and quantum gravity. Non-invertible symmetries generalize the familiar product rule of groups to a more general fusion rule, and can be gauged to form dual descriptions of invertible gauge symmetries. The authors investigate whether there are other types of non-invertible gauge symmetries, particularly in theories with gravity. They find that in theories with fundamental, tensionless strings, a new form of non-invertible gauge symmetry emerges in the limit where the string tension approaches zero. These stringy non-invertible gauge symmetries appear in standard examples such as non-abelian orbifolds, and moving away from the tensionless limit always breaks these symmetries. The paper also discusses the realization of both conventional and stringy non-invertible gauge symmetries in the AdS/CFT correspondence. Despite being generically broken, approximate non-invertible symmetries have implications for Swampland constraints, such as proving the existence of towers of states related to the Distance Conjecture and explaining the existence of slightly sub-extremal states that fill in gaps in the Sublattice Weak Gravity Conjecture. The authors provide concrete examples to illustrate their findings, including non-abelian orbifolds and toroidal orbifolds, and discuss the selection rules for non-invertible symmetries on Riemann surfaces. They conclude by highlighting the broader implications of these symmetries for string theory and quantum gravity, and suggest future directions for research.The paper explores the nature and implications of non-invertible symmetries in quantum field theory (QFT) and quantum gravity. Non-invertible symmetries generalize the familiar product rule of groups to a more general fusion rule, and can be gauged to form dual descriptions of invertible gauge symmetries. The authors investigate whether there are other types of non-invertible gauge symmetries, particularly in theories with gravity. They find that in theories with fundamental, tensionless strings, a new form of non-invertible gauge symmetry emerges in the limit where the string tension approaches zero. These stringy non-invertible gauge symmetries appear in standard examples such as non-abelian orbifolds, and moving away from the tensionless limit always breaks these symmetries. The paper also discusses the realization of both conventional and stringy non-invertible gauge symmetries in the AdS/CFT correspondence. Despite being generically broken, approximate non-invertible symmetries have implications for Swampland constraints, such as proving the existence of towers of states related to the Distance Conjecture and explaining the existence of slightly sub-extremal states that fill in gaps in the Sublattice Weak Gravity Conjecture. The authors provide concrete examples to illustrate their findings, including non-abelian orbifolds and toroidal orbifolds, and discuss the selection rules for non-invertible symmetries on Riemann surfaces. They conclude by highlighting the broader implications of these symmetries for string theory and quantum gravity, and suggest future directions for research.