The book presents the modern state of the art in the mathematical theory of compressible Navier-Stokes equations, with particular emphasis on the applications to aerodynamics. The topics covered include:modeling of compressible viscous flows; modern mathematical theory of nonhomogeneous boundary value problems for viscous gas dynamics equations;applications to optimal shape design in aerodynamics;kinetic theory for equations with oscillating data;new approach to the boundary value problems for transport equations.The monograph offers a comprehensive and self-contained introduction to recent mathematical tools designed to handle the problems arising in the theory.