We describe a new theory of sheaves of rings of differential operators on the Witt-vectors on a smooth scheme characteristic p. Roughly speaking, these sheaves are to the de Rham-Witt complex as the usual sheaf of differential operators is to the de Rham complex. Further, the module theory of these rings generalizes and extends the category of crystals; we will explain how the associated formalism gives very general comparison theorems between crystalline cohomology of a crystal and the the de Rham-Witt cohomology with values in that crystal.