Waves in linearly elastic media of infinite extent, composed of alternating layers of two different materials with perfectly bonded interfaces, exhibit some special features which are due to periodicity and impedance discontinuities at each interface. Floquet's theory permits an analysis of such waves which naturally separates into problems of anti‐plane and plane strain. The recently developed dispersion relations for such waves will be discussed, placing emphasis on the special features, such as existence of stopping bands, transition points, conical points, and asymptotic behavior. More. recent work on the applicability of Floquet‐type solutions in anti‐plane strain to layered semi‐infinite media with traction‐free surfaces parallel to the layers and to layered plates will also be discussed. A new class of surface is shown to exist which are dispersive, are restricted to a limited frequency range in some cases, and which can be propagating or evanescent. Finally. attempts to mathematically, model waves in periodically layered media by means of approximate theories will be mentioned, in particular the class of “effective dispersion” theories, which contain both the dispersive and the filtering properties.