The Navier--Stokes equations of gas- and magnetohydrodynamics are of an unusual hyperbolic--parabolic type combining features of both hyperbolic and parabolic evolution equations, and this is true both at high frequencies, corresponding to short-time behavior/well-posedness of the equations, and low frequencies, corresponding to large-time behavior. The dual nature of the equations comes to the fore in the study of stability of viscous shock waves, or pressure fronts. In this talk, we discuss the interesting blend of analytical techniques that has evolved to treat this problem, centered loosely around the spectral representation (inverse Laplace tranform) formula for C0 semigroups and the spectral determinants arising in normal modes analyses for hyperbolic and parabolic problems, known respectively as Lopatinski and Evans functions. Our results reduce the question of stability to a simple and numerically checkable {Evans function (ODE) condition}, equivalent to spectral stability plus hyperbolic stability of the associated ideal shock (easily checked) plus transverality of the connecting orbit defining the profile (automatic for extreme shocks).ngs" a subroutine