  With this work we initiate a study of the representations of a unipotent
group over a field of characteristic zero from the modular point of view. Let
$G$ be such a group. The stack of all representations of a fixed finite
dimension $n$ is badly behaved. We introduce an invariant, $w$, of $G$, its
\textit{width}, as well as a certain nondegeneracy condition on
representations, and we prove that nondegenrate representations of dimension $n
\le w+1$ form a quasi-projective variety. Our definition of the width is
opaque; as a first attempt to elucidate its behavior, we prove that it is
bounded by the length of a composition series. Finally, we study the problem of
gluing a pair of nondegenerate representations along a common subquotient.
