According to its original definition a unidirectionally electrically conducting (UEC) screen is a penetrable anisotropic surface which is perfectly conducting in one direction and perfectly insulating in the orthogonal direction. The scattering of an arbitrarily polarized plane wave obliquely incident on the edge of either a UEC half plane or a planar junction between a UEC screen and a perfectly conducting half plane is analyzed in this paper. The direction of infinite conductivity of the UEC screen is arbitrarily oriented with respect to the edge. Exact closed form expressions for the scattered field are derived, which contain the standard transition function of the uniform geometrical theory of diffraction and simple ratios of trigonometric functions. Specific attention is devoted to the phenomenon of surface wave excitation at the diffracting edge of the structures under investigation. The features of this wave guided by the anisotropic screen are discussed.