The concept of "waves in solids" usually conjures up the image of seismic waves, the modeling of which can be dated back to the works of Lord Rayleigh (1885) and many others. Amazingly, the same solutions can be used for wavelengths ranging from the kilometer scale (earthquakes) all the way to the Angstrom scale (high-frequency signal filters used in cell phones). Of course the governing equations must be modified in the process to account for anisotropy, piezoelectricity, and other effects which complicate the analysis, but also reveal surprising possibilities such as the celebrated shear-horizontal Bleustein-Gulyaev surface wave. This talk devotes its first part to an exposition of some analytical results obtained for surface waves and edge waves, which are prime examples of "linear waves in linear solids". The second part is devoted to "linearized waves in nonlinear solids", which are infinitesimal waves superimposed onto the large, static, homogeneous deformation of a hyperelastic solid. These small-amplitude motions are used to study the influence of pre-stress on the stability of geophysical structures or loaded rubber mounts, or to measure residual stresses in soft biological tissues. Finally, the talk evokes some nonlinear waves or more precisely, some exact traveling wave solutions to the equations of motion in fully nonlinear elasticity. These include generalizations of some odd "linear waves in nonlinear solids" which are of finite amplitude and yet whose propagation ends up being governed by linear differential equations. Another development is that of a simple way to account for inherent characteristic lengths in a solid, to model the dispersion of bulk waves; then, by fine-tuning dispersive and nonlinear effects, solitary and compact traveling waves can be uncovered.