Witt vectors, commutative and non-commutative

Witt vectors $W(A)$ of a commutative ring $A$ were discovered 80 years ago, but they still pop up in unexpected places and are the subject of continuous research. One question that has been solved only recently is how to generalize Witt vectors to the case when $A$ is not commutative. I am going to review the classical theory, and then show how a very natural modification leads to the non-commutative case.