  We consider complete toric varieties $X$ such that a maximal unipotent
subgroup $U$ of the automorphism group $\text{Aut}(X)$ acts on $X$ with an open
orbit. It turns out that such varieties can be characterized by several
remarkable properties. We study the set of Demazure roots of the corresponding
complete fan, describe the structure of a maximal unipotent subgroup $U$ in
$\text{Aut}(X)$, and find all regular subgroups in $U$ that act on $X$ with an
open orbit.
