  We investigate the Hermitian curvature flow (HCF) of left-invariant metrics
on complex unimodular Lie groups. We show that in this setting the flow is
governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1}
(g_t)$. The solution $g_t$ always exist for all positive times, and $(1 +
t)^{-1}g_t$ converges as $t\to \infty$ in Cheeger-Gromov sense to a non-flat
left-invariant soliton $(\bar G, \bar g)$. Moreover, up to homotheties on each
of these groups there exists at most one left-invariant soliton solution, which
is a static Hermitian metric if and only if the group is semisimple. In
particular, compact quotients of complex semisimple Lie groups yield examples
of compact non-K\"ahler manifolds with static Hermitian metrics. We also
investigate the existence of static metrics on nilpotent Lie groups and we
generalize a result in \cite{EFV} for the pluriclosed flow. In the last part of
the paper we study HCF on Lie groups with abelian complex structures.
