Topological defects in quantum field theory can be understood as a generalised notion of symmetry, where the operation is not required to be invertible. Duality transformations are an important example of this. By considering defects of various dimensions, one is naturally led to more complicated algebraic structures than just groups. So-called 2-groups are a first instance, which arise from invertible defects of codimension 1 and 2. Without invertibility one arrives at so-called "fusion categories”. These structures are already visible in dimension 2, where the field theories are under much better control. I will focus on two-dimensional conformal field theories and show how these higher structures arise.
I will explain how one can "gauge" such non-invertible symmetries of 2d CFTs, that is, define a generalised notion of an orbifold. Finally, time permitting, will briefly discuss the corresponding structures in 3d topological field theories.