  We prove that the isomorphism problem for separable nuclear C*-algebras is
complete in the class of orbit equivalence relations. In fact, already the
isomorphism of simple, separable AI C*-algebras is a complete orbit equivalence
relation. This means that any isomorphism problem arsing from a continuous
action of a separable completely metrizable group can be reduced to the
isomorphism of simple, separable AI C*-algebras. As a consequence, we get that
the isomorphism problems for separable nuclear C*-algebras and for separable
C*-algebras have the same complexity. This answers questions posed by Elliott,
Farah, Paulsen, Rosendal, Toms and T\"ornquist.
