The lecture reports joint work with T. Daskalopoulos on solutions of inverse mean curvature flow in the case of entire graphs. We first establish the global existence and uniqueness of entire graphs with super-linear growth at infinity. Then we analyse the behaviour of asymptotically conical entire convex graphs and prove as main result that the solution exists up to a finite time depending on the initial growth near infinity, when the solution converges to a flat plane. Our techniques exploit the ultra-fast diffusion character of the fully-nonlinear flow.