  We classify plethories over fields of characteristic zero, thus answering a
question of Borger-Wieland and Bergman-Hausknecht. All plethories over
characteristic zero fields are linear, in the sense that they are free
plethories on a bialgebra. For the proof we need some facts from the theory of
ring schemes where we extend previously known results. We also classify
plethories with trivial Verschiebung over a perfect field of non-zero
characteristic and indicate future work.
