This work provides a unified formalism for studying difference and (Hasse-)
differential algebraic geometry, by introducing a theory of "iterative Hasse
rings and schemes". As an application, Hasse jet spaces are constructed
generally, allowing the development of the theory for arbitrary systems of
algebraic partial difference/differential equations, where constructions by
earlier authors applied only to the finite dimensional case. In particular, it
is shown that under appropriate separability assumptions a Hasse variety is
determined by its jet spaces at a point.