  We develop the theory of higher prolongations of algebraic varieties over
fields in arbitrary characteristic with commuting Hasse-Schmidt derivations.
Prolongations were introduced by Buium in the context of fields of
characteristic 0 with a single derivation. Inspired by work of Vojta, we give a
new construction of higher prolongations in a more general context.
Generalizing a result of Buium in characteristic 0, we prove that these
prolongations are represented by a certain functor, which shows that they can
be viewed as `twisted jet spaces.' We give a new proof of a theorem of Moosa,
Pillay, and Scanlon that the prolongation functor and jet space functor
commute. We also prove that the $m^{th}$-prolongation and $m^{th}$-jet space of
a variety are differentially isomorphic by showing that their infinite
prolongations are isomorphic as schemes.
