Singular hermitian metrics for holomorphic line bundles have long played a leading role in many areas of complex analysis and geometry. By contrast, the higher rank case is relatively new, in part because there are difficulties in trying to define the curvature. A breakthrough originating in the work of Berndtsson and Paun is to abandon the curvature itself, and instead try to define a notion of positive curvature. They succeeded in defining Griffiths positivity, but Nakano positivity remained illusive. Since then, notions of Nakano positivity have been introduced, but their flavor is more based in PDE than Geometry. I will present and discuss a notion that I recently proposed, as well as some applications.