A geometrical theory is devised for the description and calculation of surface waves, such as Rayleigh waves, on boundaries or interfaces of elastic solids. It applies to curved surfaces and to inhomogeneous media. The theory, which is an extension of geometrical optics, involves complex rays that travel from the source to the surface, then along the surface, and finally from the surface to points in the solid. It also includes phases and amplitudes associated with each point on each ray. Geometrical formulas are derived for the determination of these phases and amplitudes. They involve certain excitation and radiation coefficients, which are also determined. The total displacement at a point is the sum of the displacements on all the rays through the point, each of which is constructed from the corresponding phase and amplitude. The theory applies to periodic waves of high frequency and short wavelength, as well as to the rapidly varying portions of any waveform. It is a generalization of the authors' earlier geometrical theory of scalar surface waves [J. Appl. Phys.31, 1039 (1960)].