This short contribution to the Gates Festschrift on ''Surface Science:
The First Thirty Years'' describes how a simple general theory develo
ped. It is the story of the formation of a useful mathematical languag
e for the study of surfaces, interfaces, superlattices and composite m
aterials. Personal accounts of the evolution of the theory are given a
nd illustrations come mostly from the field of vibrations. This paper
tries also to emphasize the benefits wrought by such a fundamental res
earch.