2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/96441We review some recent developments of Grad's approach to solving the Boltzmann equation and creating reduced description. The method of invariant manifold is put forward as a unified principle to establish corrections to Grad's equations. A consistent derivation of regularized Grad's equations in the framework the method of invariant manifold is given. A new class of kinetic models to lift the finite-moment description to a kinetic theory in the whole space is established. Relations of Grad's approach to modern mesoscopic integrators such as the entropic lattice Boltzmann method are also discussed.35 pages, a review paper, Continuum Mechanics and Thermodynamics, in pressStatistical Mechanicstext