Anisotropy means that the physical properties of a solid medium vary with direction.For example, the speed of travel of a wave in the vertical direction may be different from its speed in the horizontal direction. The real subsurface of the Earth is anisotropic in certain regions. Media containing fractures can be effectively modeled by replacing them with a single anisotropic medium. This chapter introduces the reader to the mathematical theory of how seismic waves propagate in anisotropic media.Isotropic media can be described by two physical parameters, whereas anisotropic media require more. The relatively simple but important case of a transversely isotropic medium, which requires five physical parameters or elastic constants for its description, is covered in relative detail.Transversely isotropic media that are weakly anisotropic are also discussed, as well as special cases, such as elliptical anisotropy. Reflection and transmission are briefly discussed. Slowness surfaces, which are helpful in understanding the nature of anisotropy in a medium, are discussed. The interesting effects that occur in a tilted anisotropic medium are also briefly covered.