Non-commutative Witt vectors

Witt vectors were originally defined for a commutative ring $A$. About 15 years ago L. Hesselholt discovered a generalization of this construction to non-commutative rings. His approach was heavily based on non-trivial facts from stable homotopy theory. I will present an elementary re-interpretation of Hesselholt's results which also allows to strengthen and clarify them. If time permits, I will also touch upon some applications.