In this talk we will briefly review   the mathematical theories on the compressible viscous fluid developed in terms of pointwise structure of the wave propagators. The mathematical compressible viscous fluids include conservation laws with artificial viscosity, compressible Navier-Stokes equations, Boltzmann equations, and Lax-Friedrichs scheme. One interest is on constructing the wave propagators around the far fields and use it to construct the nonlinear waves scattering theory over a shock layer for planar wave perturbation; and one can use the wave propagators to construct the invariant manifolds for the Boltzmann equation and study the bifurcation problems of the condensation evaporation problem for the Boltzmann equation in a half space domain.