This book is concerned with the methods of solving the nlinear Boltz- mann equation and of investigating its possibilities for describing some aerodynamic and physical problems. This mograph is a sequel to the book 'Numerical direct solutions of the kinetic Boltzmann equation' (in Russian) which was written with F. G. Tcheremissine and published by the Computing Center of the Russian Academy of Sciences some years ago. The main purposes of these two books are almost similar, namely, the study of nequilibrium gas flows on the basis of direct integration of the kinetic equations. Nevertheless, there are some new aspects in the way this topic is treated in the present mograph. In particular, attention is paid to the advantages of the Boltzmann equation as a tool for considering nequi- librium, nlinear processes. New fields of application of the Boltzmann equation are also described. Solutions of some problems are obtained with higher accuracy. Numerical procedures, such as parallel computing, are in- vestigated for the first time. The structure and the contents of the present book have some com- mon features with the mograph mentioned above, although there are new issues concerning the mathematical apparatus developed so that the Boltzmann equation can be applied for new physical problems. Because of this some chapters have been rewritten and checked again and some new chapters have been added.