A powerful tool to study the geometry of Radon measures is the decomposability bundle, which I introduced with Alberti in [On the differentiability of Lipschitz functions with respect to measures in the Euclidean space, GAFA, 2016]. This is a map which, roughly speaking, captures at almost every point the tangential directions to the Lipschitz curves along which the measure can be disintegrated. In this talk I will discuss some recent applications of this flexible tool, including a characterization of rectifiable measures as those measures for which Lipschitz functions admit a Lusin type approximation with functions of class ${C^1}$, the converse of Pansu's theorem on the differentiability of Lipschitz functions between Carnot groups, and a characterization of Federer-Fleming flat chains with finite mass.