The Boltzmann equation is an integro-differential equation whose integral kernel has the singularity in the collisional angle between two particles, in physically important models. It is known that many natural properties of the Boltzmann equation are preserved in spite of Grad's angular cutoff approximation to avoid this singularity. However, there is one important property lost after this cutoff approximation, that is, the collision integral operator without angular cutoff intrinsically behaves like the differential operator, and hence it has been expected that the Boltzmann equation enjoys the regularizing effect of solutions, such as the heat equation and the Kolmogorov equation, corresponding to the spatially homogeneous case and the inhomogeneous one, respectively. In this talk, the smoothing effect and the time asymptotic state of solutions to the Cauchy problem with the measure-valued initial data will be firstly considered for the spatially homogeneous Boltzmann equation of the Maxwellian molecule type, and, secondly, the global well-posedness under the perturbation frame work around the equilibrium will be discussed in spatially critical Besov spaces for the Boltzmann equation of hard potentials. The former part of this talk is based on joint works with T. Yang, S. Wang and H. Zhao, and the latter is with S. Sakamoto.