  We study the Hermitian curvature flow of locally homogeneous non-K\"ahler
metrics on compact complex surfaces. In particular, we characterize the
long-time behavior of the solutions to the flow. We also provide the first
example of a compact complex non-K\"ahler manifold admitting a finite time
singularity for the Hermitian curvature flow. Finally, we compute the
Gromov-Hausdorff limit of immortal solutions after a suitable normalization.
Our results follow by a case-by-case analysis of the flow on each complex model
geometry.
