The Bobylev approach to the nonlinear Boltzmann equation is reviewed. The linearized problem is discussed and it is shown that eigenfunctions decaying like a negative power of the velocity are possible with Maxwell molecules only. The relaxation to equilibrium according to the nonlinear equation is discussed and the Krook-Wu conjecture on the status of the BKW mode is shown to be false in general. The buildup of the high-energy tails is considered and a phenomenon observed by Tjon is given a simple explanation. Finally, the method is illustrated with numerical calculations performed for two sets of initial conditions.