The outstanding points of our book consist of investigations into the possibility of the numerical schemes of the direct method for solving the Boltzmann equation. Both deterministic and Monte Carlo procedures are considered to evaluate the collision integrals. The main mathematical tool is the conservative splitting method on the basis of which, a set of classical and new problems are solved to study nonequilibrium gas flows. This monograph differs from other books in the same field, because, for example the book by G.A. Bird is concerned with the approach of simulation of rarefied gas flows and the book by C. Cercignani deals with the classical kinetic theory issues and describes mainly the analytical and engineering methods for solving the Boltzmann equation. Our book is the first (as we know) monograph which is devoted to the numerical direct solving of the Boltzmann equation. The intended level of readership are graduate and postgraduate students and researches. This book can be used by the target groups as the mathematical apparatus to numerical study of complex problems of nonequilibrium gas flows.