  Given $\alpha\in(0,1]$ and $p\in[1,+\infty]$, we define the space
$\mathscr{DM}^{\alpha,p}(\mathbb R^n)$ of $L^p$ vector fields whose
$\alpha$-divergence is a finite Radon measure, extending the theory of
divergence-measure vector fields to the distributional fractional setting. Our
main results concern the absolute continuity properties of the
$\alpha$-divergence-measure with respect to the Hausdorff measure and
fractional analogues of the Leibniz rule and the Gauss-Green formula. The
sharpness of our results is discussed via some explicit examples.
