This paper describes the mathematical models derived for wave propagation in solids
with internal structure. The focus of the overview is on one-dimensional models
which enlarge the classical wave equation by higher-order terms. The crucial
parameter in models is the ratio of characteristic lengths of the excitation and the
internal structure. Novel approaches based on the concept of internal variables permit
one to take the thermodynamical conditions into account directly. Examples of
generalisations include frequency-dependent multiscale models, nonlinear models and
thermoelasticity. The substructural complexity within the framework of elasticity
gives rise to dispersion of waves. Dispersion analysis shows that acoustic and optical
branches of dispersion curves together describe properly wave phenomena in
microstructured solids. In the case of nonlinear models, the governing equations are
of the Boussinesq type. It is argued that such models of waves in solids with
microstructure display properties that can be analysed as phenomena of
complexity.