In this talk, I will present recent results in collaboration with Esther Cabezas-Rivas and Marcos Solera (U. Valencia) concerning the analogous to the inverse mean curvature flow in the presence of an anisotropy (including the case of crystalline anisotropies). I will show the construction of weak solutions of the anisotropic inverse mean curvature flow under very mild assumptions both on the anisotropy and on the initial data. The existence is shown by means of an approximation procedure with anisotropic p-harmonic functions. Our notion of weak solution, which turn out to be unique, still recovers variational and geometric properties similar to those introduced by Huisken-Ilmanen in the classical isotropic setting. We also reach classical results like continuity or exponential growth of the perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition, the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, I will show some examples of explicit solutions