Like many conservation laws, the compressible Euler equations readily form shocks from smooth data. However, their viscous counterparts like Navier–Stokes typically prevent such singularities. This minicourse will focus on the interface between these behaviors: the vanishing-viscosity limit near shock formation. Beginning from simple calculations, we will develop a matched asymptotic expansion bridging hyperbolic and parabolic dynamics. This allows us to identify sharp vanishing-viscosity rates in Hölder norms and a universal viscous layer near shock formation. The lectures represent joint work with John Anderson and Sanchit Chaturvedi.