The aim of this talk is to provide an overview of recent results, obtained jointly with Graziano Crasta and Virginia De Cicco, regarding the chain rule for the distributional divergence of composite functions $v(x) = B(x, u(x))$, where $B(\cdot, t)$ is a bounded divergence-measure vector field and $u$ is a scalar function of bounded variation. To ensure the validity of the divergence rule in this irregular setting, we introduce a family of nonlinear pairings. The elements of this family depend crucially on the choice of the pointwise representative of $u$ on its jump set. I will discuss key structural properties of these pairings, such as coarea formulas, integral representations, and the Gauss-Green formulas.  Furthermore, I will address the variational implications of this theory by characterizing the pairings that ensure lower semicontinuity of the corresponding functionals  w.r.t. strict convergence in $BV$. Finally, we will see that these pairing measures can be variationally regarded as the relaxation of integral functionals originally defined in Sobolev spaces.