  Non-invertible symmetries in quantum field theory (QFT) generalize the
familiar product rule of groups to a more general fusion rule. In many cases,
gauged versions of these symmetries can be regarded as dual descriptions of
invertible gauge symmetries. One may ask: are there any other types of
non-invertible gauge symmetries? In theories with gravity we find a new form of
non-invertible gauge symmetry that emerges in the limit of fundamental,
tensionless strings. These stringy non-invertible gauge symmetries appear in
standard examples such as non-abelian orbifolds. Moving away from the
tensionless limit always breaks these symmetries. We also find that both the
conventional form of non-invertible gauge symmetries and these stringy
generalizations are realized in AdS/CFT. Although generically broken,
approximate non-invertible symmetries have implications for Swampland
constraints: in certain cases they can be used to prove the existence of towers
of states related to the Distance Conjecture, and can sometimes explain the
existence of slightly sub-extremal states which fill in the gaps in the
sublattice Weak Gravity Conjecture.
