  Orbifolds of two-dimensional quantum field theories have a natural
formulation in terms of defects or domain walls. This perspective allows for a
rich generalisation of the orbifolding procedure, which we study in detail for
the case of topological field theories. Namely, a TFT with defects gives rise
to a pivotal bicategory of "worldsheet phases" and defects between them. We
develop a general framework which takes such a bicategory B as input and
returns its "orbifold completion" B_orb. The completion satisfies the natural
properties B \subset B_orb and (B_orb)_orb = B_orb, and it gives rise to
various new equivalences and nondegeneracy results. When applied to TFTs, the
objects in B_orb correspond to generalised orbifolds of the theories in B. In
the example of Landau-Ginzburg models we recover and unify conventional
equivariant matrix factorisations, prove when and how (generalised) orbifolds
again produce open/closed TFTs, and give nontrivial examples of new orbifold
equivalences.
