Unipotent group schemes over integral rings

In this paper we study families of unipotent algebraic groups over integral rings. The main results relate to the geometry of such families. In particular, we prove that, under some hypotheses, the space of such a family is isomorphic to an affine space over the base. We give counterexamples showing that in the case of an arbitrary base ring the basic facts of the theory of unipotent algebraic groups over a field cease to be true. For a certain class of the group schemes that we consider we prove results on cohomology, extensions and deformations.