Two-dimensional topological field theories have three layers of structure: bulk theories living on two-dimensional worldsheet patches, one-dimensional defect lines between such patches, and zero-dimensional field insertions. We review how these layers naturally combine into the structure of a bicategory, and sketch how sigma models or Landau–Ginzburg models provide relevant examples. Next we explain how to encode orbifolds in entirely algebraic language in this general setting. In fact, this leads to a generalisation of orbifolds (and in this sense of "symmetry"), and to a construction of new equivalences of field theories. Much of this is based on joint work with Ingo Runkel.