This study deals with several two-dimensional scattering and diffraction problems in anisotropic media. The intent is twofold: First, to generalize mathematical methods applicable in isotropic regions to a certain class of anisotropic problems; and Second, to study the solutions of the anisotropic problems in such a manner as to highlight certain common properties which point the way toward the construction of approximate solutions for configurations with more general structural shape or anisotropy. In the low-frequency range, the method of multipole expansion is used. It is demonstrated that the rigorous solution for the problem of scattering of a plane or cylindrical wave by an obstacle which is small compared to the wavelength may be expanded in a series whose terms correspond to multipole radiation in the anisotropic medium. As an illustration, the excitation coefficients of the first few terms arising from scattering by a narrow conducting ribbon are calculated. In the high frequency range, geometrical optics, the first-order asymptotic solution of Maxwell's equations is considered first. The ray refractive index is calculated, and the laws of propagation and reflection of rays, which define the trajectories of energy transport, are derived from Fermat's principle. To obtain an insight into diffraction phenomena, two types of representative problems --- diffraction by a straight edge, and diffraction by a smoothly curved object -- are discussed.