In this talk, I will review briefly the classical Prandtl boundary layer asymptotic expansions in the study of structure of viscous fluids under the high Reynolds number in a domain with boundaries. And then I will discuss the vanishing viscosity limit of solution to initial boundary value problem for 2D compressible magneto-hydrodynamics (MHD) equations in Sobolev spaces, where the velocity is always imposed the no-slip boundary condition. The effects of the normal component of magnetic field, tangential component of magnetic field and magnetic diffusion on the mathematical analysis for stability of MHD boundary layer and vanishing viscosity limit of solution to compressible MHD equations will be addressed in details.