In the first half of the talk I want to give an introduction to the classical story of Witt vectors over commutative rings. In the second half, I want to discuss a generalisation of Hesselholt's noncommutative Witt vectors from recent work with Dotto, Nikolaus and Patchkoria. This is motivated by a connection to trace methods in algebraic K-theory, but also has a couple of interesting algebraic consequences, including a generalisation of characteristic polynomials to noncommutative rings.