  We develop a general theory of 3-dimensional ``orbifold completion'', to
describe (generalised) orbifolds of topological quantum field theories as well
as all their defects. Given a semistrict 3-category $\mathcal{T}$ with adjoints
for all 1- and 2-morphisms (more precisely, a Gray category with duals), we
construct the 3-category $\mathcal{T}_{\textrm{orb}}$ as a Morita category of
certain $E_1$-algebras in $\mathcal{T}$ which encode triangulation invariance.
We prove that in $\mathcal{T}_{\textrm{orb}}$ again all 1- and 2-morphisms have
adjoints, that it contains $\mathcal{T}$ as a full subcategory, and we argue,
but do not prove, that it satisfies a universal property which implies
$(\mathcal{T}_{\textrm{orb}})_{\textrm{orb}} \cong \mathcal{T}_{\textrm{orb}}$.
This is a categorification of the work in [CR]. Orbifold completion by design
allows us to lift the orbifold construction from closed TQFT to the much richer
world of defect TQFTs. We illustrate this by constructing a universal
3-dimensional state sum model with all defects from first principles, and we
explain how recent work on defects between Witt equivalent Reshetikhin--Turaev
theories naturally appears as a special case of orbifold completion.
