  In the last three decades, Fourier analysis methods have known a growing
importance in the study of linear and nonlinear PDE's. In particular,
techniques based on Littlewood-Paley decomposition and paradifferential
calculus have proved to be very efficient for investigating evolutionary fluid
mechanics equations in the whole space or in the torus. We here give an
overview of results that we can get by Fourier analysis and paradifferential
calculus, for the compressible Navier-Stokes equations. We focus on the Initial
Value Problem in the case where the fluid domain is the whole space or the
torus in dimension at least two, and also establish some asymptotic properties
of global small solutions. The time decay estimates in the critical regularity
framework that are stated at the end of the survey are new, to the best of our
knowledge.
