Pre-stressed bodies are ubiquitous in technical applications and in several fields of science. For instance in biomechanics, the mechanical behaviour in service of many soft biological tissues (such as arterial walls, veins, skin, tendons, etc.) can be explained by modelling them as pre-stressed viscoelastic materials. In civil engineering, bridge bearings or seismic shock absorbers under a building are clear examples of devices operating in conditions of pre-stress, and sometimes subject to dramatically large strains. Other examples can be found in the automotive industry, in seismology, in oil prospecting, in non-destructive ultrasonic evaluation, in high frequency signal processing for electronic devices, in fibers optics, etc.The study of wave motion is a natural and revealing approach to the proper- ties of a pre-stressed body. Indeed, acoustic waves may be used to evaluate the material parameters of a given elastic body or, if these are known, to evaluate the state of induced anisotropy or of residual stress (in fact, they may well be the only way to evaluate the mechanical characteristics of soft tissues in vivo). They may also be used to detect structural defects. Other major interests include the study of standing waves, with applications to stability and bifurcation analyses, and the study of nonlinear waves, with applications to shock formations and solitary waves generation. Hence, the understanding of the mathematics and of the mechanics of dynamical problems in pre-stressed elastic and viscoelastic materials is of paramount importance to many applications. Nevertheless, a recent comprehensive synthetic textbook is lacking in this field.The aim of these lecture notes is to take a first step toward the eventual elaboration of such a reference volume, by providing a unique, state-of-the-art, multi- disciplinary overview on the subject of linear, linearized, and nonlinear waves in pre-stressed materials. This is achieved through the interaction of several topics, ranging from the mathematical modelling of incremental material elastic response, to the analysis of the governing differential equations and related boundary-value problems, and to computational methods for the numerical solution to these problems, with particular reference to industrial, geophysical, and biomechanical applications. We have tried to achieve this goal by including:• A unified introduction to wave propagation (small-on-large and large-on- large);• The basic and fundamental theoretical issues (mechanical modelling, exact solutions, asymptotic methods, numerical treatment);• A perspective on classical (such as geophysics), current (such as the mechanics of rubber-like solids), and emergent (such as nonlinear solid biomechanics) applications.