Biaxial -medium is an artifical material which is obtained by diffusion of planar conducting microstructures, having the shape of , into an isotropic dielectric medium with suitable orientations. In this paper, based on the eigen plane wave spectrum representation of the field and the Fourier expansion for the unknown angular spectrum amplitude, the cylindrical vcctor-wave-function representations of the electromagnetic fields in such materials are developed. It is shown that the solutions of the source-free Maxwell's equations for a biaxial -medium are composed of two cigenwaves traveling with different wave numbers, and each eigenwave is a superposition of two transverse waves and a longitudinal wave. The addition theorem of wave functions for biaxial media can be derived from that of wave functions for isotropic media. Applications of the theory are made to the case of two-dimensional scattering of a plane wave by a biaxial circular cylinder. Numerical results for some cases are presented.