Parabolic Harnack inequalities were proved by Moser for linear equation with bounded and measurable coefficients. In the case of the parabolic p-Laplacean such kind of estimates cannot hold (as proved by Trudinger). In the nineties DiBenedetto introduced the so called intrinsic Harnack inequalities for the protype equation. His original proof requiries the maximum principle and the existence of suitable subsolutions. Therefore the proof for general equations (with bounded and measurable coefficient) was missing. In some recent papers, in collaboration with DiBendetto and Gianazza, we proved intrinsic Harnack inequalities for general degenerate and singular operators. We show, via suitable counterexamples, that such estimates are sharp. Moreover we proved that when p is approaching to 2, our estimates tend to the classical Moser estimates.