Witt vectors are an important construction in number theory, generalizing the passage from \(\mathbb F_p\) to \(\mathbb Z/p^k\) and \(\mathbb Z_p\). Interestingly, this construction, and others related to it, also arises naturally out of the formalism of \(S^1\)-equivariant stable homotopy theory—in particular the theory surrounding algebraic \(K\)-theory and the cyclotomic trace map. I will sketch a picture and some intuitions for the many connections between arithmetic geometry and topological cyclic homology, and then explain some of my recent work in this area.