  Let $G$ be an affine algebraic group scheme over a field $k$. We show there
exists a unipotent normal subgroup of $G$ which contains all other such
subgroups; we call it the restricted unipotent radical $\mathrm{Rad}_u(G)$ of
$G$. We investigate some properties of $\mathrm{Rad}_u(G)$, and study those $G$
for which $\mathrm{Rad}_u(G)$ is trivial. In particular, we relate these
notions to their well-known analogues for smooth connected affine $k$-groups.
