High frequency homogenisation is applied to develop asymptotics for waves propagating along interfaces or structured surfaces. The asymptotic method is a two-scale approach fashioned to encapsulate the microstructural information in an effective homogenised macro- scale model. These macroscale continuum representations are constructed to give solutions near standing wave frequencies and are valid even at high frequencies. The asymptotic the- ory is adapted to model dynamic phenomena in functionally graded waveguides and in periodic media, revealing their similarities. Demonstrating the potential of high frequency homogenisation, the theory is extended for treating localisation phenomena in discrete peri- odic media containing localised defects, and for identifying Rayleigh-Bloch waves. In each of the studies presented here the asymptotics are complemented by analytical or numerical solutions, or both.