  We present some recent developments on shock capturing methods for nonlinear
hyperbolic systems of balance laws, whose prototype is the Euler system of
compressible fluid flows, and especially discuss {structure-preserving}
techniques. The problems under consideration arise with complex fluids in
realistic applications when friction terms, geometrical terms, viscosity and
capillarity effects, etc., need to be taken into account in order to achieve a
proper description of the physical phenomena. For these problems, it is
necessary to design numerical methods that are not only consistent with the
given partial differential equations, but remain accurate and robust in certain
{asymptotic regimes} of physical interest. That is, certain structural
properties of these hyperbolic problems (conservation or balance law,
equilibrium state, monotonicity properties, etc.) are essential in many
applications, and one seeks that the numerical solutions preserve these
properties, which is often a very challenging task.
