Classical results in ergodic theory due to Dye and Ornstein--Weiss show that, for an arbitrary countable amenable group, any two free ergodic measure-preserving actions on the standard atomless probability space are orbit equivalent, i.e. their orbit equivalence relations are isomorphic. This motivates the question of what happens for nonamenable groups. Works of Ioana and Epstein showed that, for an arbitrary countable amenable group, the relation of orbit equivalence of free ergodic measure-preserving actions on the standard probability space has uncountably many classes. In joint work with Gardella, we strengthen these conclusions by showing that such a relation is in fact not Borel. The proof makes essential use of techniques from operator algebras, including cocycle superrigidity results due to Popa, and answers a question of Kechris.