On the grazing collision limit for the spatially homogeneus Boltzmann equation

We review here some recent developments connected to the asymptotics of the spatially homoheneus Boltzmann equation when the collision become grazing. It has been shown recently by Villani that, for a very broad class of potentials, solutions of the Boltzmann equation converge weakly to solutions of the Landau equation. This asymptotic result can be improved for the non cut-off Kac equation and for the Boltzmann equation with Maxwell molecules.