Suppose we have a holomorphic differential form, defined on the smooth locus of a complex space. Under what conditions does it extend to a holomorphic differential form on a resolution of singularities? In 2011, Greb, Kebekus, Kovacs, and Peternell proved that such an extension always exists on algebraic varieties with klt singularities. I will explain how to solve this problem in general, with the help of Hodge modules and the Decomposition Theorem. This is joint work with Kebekus.