This review covers the propagation of acoustic surface waves (ASW) in periodic structures. The authors discuss the main types of ASW: Rayleigh waves, Gulyaev--Bleustein waves, Love waves. Shear surface waves which can propagate along a periodically uneven surface of an elastic solid are described in detail. The authors rigorously treat the Bragg reflection of ASW from periodic arrays of grooves on the substrate for the cases of normal, oblique, and side incidence. Reciprocal conversion effects between bulk and surface waves in an array whose period is close to the ASW wavelength are discussed, together with such related effects as the interaction between Rayleigh waves and Lamb modes, acoustic wave transmission through a gap in a piezoelectric material, etc. Laser excitation of ASW and magnetostatic wave propagation in periodic structures are also reviewed.