Differently from their integer versions, the fractional Sobolev spaces $W^{\alpha,p}(R^n)$ do not seem to have a clear distributional nature. By exploiting suitable notions of fractional gradient and divergence already existing in the literature, we introduce via a distributional approach the new spaces $BV^{\alpha,p}(R^n)$ of $L^p$ functions with bounded $\alpha$-fractional variation in $R^n$, for $\alpha \in (0,1)$ and $p \in [1, \infty]$. In addition, we define in a similar way the distributional fractional Sobolev spaces $S^{\alpha,p}(R^n)$, which extend naturally $W^{\alpha,p}(R^n)$. The properties of these spaces have been analyzed in a series of papers in collaboration with Giorgio Stefani, Elia Bruè, Mattia Calzi and Daniel Spector. In this talk, we shall focus ourselves on the absolute continuity property of the fractional variation with respect to a suitable Hausdorff measure, and subsequently on the fractional Leibniz rules involving $BV^{\alpha,p}$ and $S^{\alpha,p}$ functions. As an application of these results we will show the well-posedness of the boundary-value problem for a general $2\alpha$-order fractional elliptic operator in divergence form.