The book presents the modern state of the art in the mathematical theory of compressible Navier-Stokes equations, with particular emphasis on 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.