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Interfaces advancing through random media represent a number of differentproblems in physics, biology and other disciplines. Here, we study thepinning/depinning transition of the prototypical non-equilibrium interfacialmodel, i.e. the Kardar-Parisi-Zhang equation, advancing in a disordered medium.We analyze separately the cases of positive and negative non-linearitycoefficients, which are believed to exhibit qualitatively different behavior:the positive case shows a continuous transition that can be related todirected-percolation-depinning while in the negative case there is adiscontinuous transition and faceted interfaces appear. Some studies haveargued from different perspectives that both cases share the same universalbehavior. Here, by using a number of computational and scaling techniques weshed light on this puzzling situation and conclude that the two cases areintrinsically different.

Interfacial depinning transitions in disordered media: revisiting an old puzzle