  We summarize recent progress on one- and multi-dimensional stability of
viscous shock wave solutions of compressible Navier--Stokes equations and
related symmetrizable hyperbolic--parabolic systems, with an emphasis on the
large-amplitude regime where transition from stability to instability may be
expected to occur. The main result is the establishment of rigorous necessary
and sufficient conditions for linearized and nonlinear planar viscous
stability, agreeing in one dimension and separated in multi-dimensions by a
codimension one set, that both extend and sharpen the formal conditions of
structural and dynamical stability found in classical physical literature. The
sufficient condition in multi-dimensions is new, and represents the main
mathematical contribution of this article. The sufficient condition for
stability is always satisfied for sufficiently small-amplitude shocks, while
the necessary condition is known to fail under certain circumstances for
sufficiently large-amplitude shocks; both are readily evaluable numerically.
The precise conditions under and the nature in which transition from stability
to instability occurs are outstanding open questions in the theory.
