Title: Minimal currents and the Abundance conjecture Abstract: The Abundance conjecture predicts that on a minimal projective klt pair (X,D), the adjoint divisor K_X+D is semiample. In this talk, I will explain how currents, and in particular currents with minimal singularities, play a role in recent progress towards a proof of the Abundance conjecture for minimal klt pairs with non-vanishing Euler-Poincaré characteristic.