  We consider surfaces with parallel mean curvature vector field and finite
total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$,
where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these
surfaces. When $n=2$, our results are similar to those obtained in \cite{bds}
for surfaces with constant mean curvature in space forms.
