  By employing the differential structure recently developed by N. Gigli, we
first give a notion of functions of bounded variation ($BV$) in terms of
suitable vector fields on a complete and separable metric measure space
$(\mathbb{X},d,\mu)$ equipped with a non-negative Radon measure $\mu$ finite on
bounded sets. Then, we extend the concept of divergence-measure vector fields
$\mathcal{DM}^p(\mathbb{X})$ for any $p\in[1,\infty]$ and, by simply requiring
in addition that the metric space is locally compact, we determine an
appropriate class of domains for which it is possible to obtain a Gauss-Green
formula in terms of the normal trace of a $\mathcal{DM}^\infty(\mathbb{X})$
vector field. This differential machinery is also the natural framework to
specialize our analysis for ${\mathsf{RCD}(K,\infty)}$ spaces, where we exploit
the underlying geometry to determine the Leibniz rules for
$\mathcal{DM}^\infty(\mathbb{X})$ and ultimately to extend our discussion on
the Gauss-Green formulas.
