A procedure is developed for finding asymptotic expansions at high frequencies of the solutions of Helmholtz's equation subject to boundary conditions on certain guiding surfaces. This includes surface waves along surfaces of rather general shapes, and wave-guide modes in a class of non-uniform waveguides. Guided waves have some features of both eigenvalue (mode) and radiation problems. The method of this paper combines the two techniques, finding “modes” that propagate along rays in the general waveguide region and whose amplitudes vary along the paths of propagation. The phases of these modes are found from two coupled equations, one analogous to the eiconal equation of geometrical optics, and the other analogous to the eigenvalue or “transverse resonance” equation of waveguides. The amplitudes are asymptotic series in inverse powers of the wavenumber, and the coefficients satisfy a set of ordinary differential equations that can be solved recursively. It is found that the ray paths are not only functions of the refractive index (as in “pure” radiation problems), but depend also on the local geometrical properties of the guiding surface.