Wave propagation is any of the ways in which waves travel. With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves. For electromagnetic waves, propagation may occur in a vacuum as well as in a material medium. Other wave types cannot propagate through a vacuum and need a transmission medium to exist. Mechanical waves are waves which propagate through a material medium (solid, liquid, or gas) at a wave speed which depends on the elastic and inertial properties of that medium. The book is intended to emphasise and to compare the various aspects of interest which include the necessary mathematical background, constitutive theories for material of differential type, polarised and shock waves, and second sound in solids at low temperatures. This book is focused on acoustic waves in fluid media and elastic perturbations in heterogeneous media. Many different systems are analysed in this book like layered media, solitons, piezoelectric substrates, crystalline systems, granular materials, interface waves, phononic crystals, acoustic levitation and soft media.