In this talk, after reviewing the work on global well-posednessof the Boltzmann equation without angular cutoff with algebraic decay tails,we will present a recent work on the global weighted L∞-solutions to the Boltzmann equation without angular cut off in the regime close to equilib-rium. A De Giorgi type argument, well developed for diffusion equations, iscrafted in this kinetic context with the help of the averaging lemma. Mores pecifically, we use a strong averaging lemma to obtain suitable Lp estimates for level-set functions. These estimates are crucial for constructing an ap-propriate energy functional to carry out the De Giorgi argument. Then weextend local solutions to global by using the spectral gap of the linearized Boltzmann operator with the convergence to the equilibrium state obtainedas a byproduct. This result fill in the gap of well-posedness theory for the Boltzmann equation without angular cut off in the L∞framework. The talk is based on the joint works with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun.