Abstract: A linear algebraic group is called a Cayley group if it is equivariantly birationally isomorphic to its Lie algebra. It is stably Cayley if the product of the group and some torus is Cayley. Cayley gave the first examples of Cayley groups with his Cayley map back in 1846. In joint work with Blunk, Borovoi, Kunyavskii and Reichstein, we classify the simple stably Cayley groups over an arbitrary field of characteristic $0$.