  We provide a `user guide' to the literature of the past twenty years
concerning the modeling and approximation of discontinuous solutions to
nonlinear hyperbolic systems that admit small-scale dependent shock waves. We
cover several classes of problems and solutions: nonclassical undercompressive
shocks, hyperbolic systems in nonconservative form, boundary layer problems. We
review the relevant models arising in continuum physics and describe the
numerical methods that have been proposed to capture small-scale dependent
solutions. In agreement with the general well-posedness theory, small-scale
dependent solutions are characterized by a kinetic relation, a family of paths,
or an admissible boundary set. We provide a review of numerical methods (front
tracking schemes, finite difference schemes, finite volume schemes), which, at
the discrete level, reproduce the effect of the physically-meaningful
dissipation mechanisms of interest in the applications. An essential role is
played by the equivalent equation associated with discrete schemes, which is
found to be relevant even for solutions containing shock waves.
