The method of asymptotic solution of waveguide equations for higher-order modes is generalized to the three-dimensional case. Geometrical optics and full wave-type solutions are constructed in the form of integral representations. In the geometrical optics case, two sorts of partial waves are used to construct the integral representation of the field: ray-type waves and Gaussian beam-type waves. In full wave solution the elementary waves of the integral representation are constructed by means of complex phase in Fresnel diffraction approximation for forward scattering. Asymptotic integral representations developed take into account coupling of the modes, describe the multiray effects and focusing of the higher-order mode field. The method is valid for waveguides with either varying height or varying boundary conditions and dielectric properties.