Since the 80's, Fourier analysis methods have known a growing importance in the study of linear and nonlinear PDE's. In particular, after the seminal paper by J.-M. Bony [Ann. Sci. École Norm. Sup. 14 (1981)], the use of Littlewood-Paley decomposition and paradifferential calculus brought a number of new results whenever those equations are considered in the whole space or the torus. We here aim at giving a survey of recent advances obtained by Fourier analysis methods in the context of compressible fluid mechanics. For simplicity, we shall focus on the barotropic compressible Navier-Stokes equations.