Motivated by topological quantum field theory, one can start from any pivotal bicategory B and construct its "orbifold completion" B_orb in terms of certain Frobenius algebras. 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. I will explain this construction (which is joint work with Ingo Runkel) and apply it to the bicategory of isolated singularities and matrix factorisations. Applications include a unified perspective on ordinary equivariant matrix factorisations, a one-line proof of Knörrer periodicity, and new equivalences for simple singularities.