From the introduction: First the author obtains a simplified expression for the Landau equation with Maxwellian molecules; after a brief discussion of the special case of isotropic distributions, where explicit solutions are easily available, he studies the form of the collision operator in the general case, then turns to the Cauchy problem associated to the equation, and insists on its regularizing properties. This leads him to rewrite the Landau collision operator as the sum of several operators. Finally, some qualitative features of the solutions, as the decay to equilibrium, the temperature tails and the positivity are given and a family of particular self-similar solutions is exhibited.