Ray theory is the cornerstone of high-frequency, body-wave seismology. Without it, seismic signals in realistic, complex media would be extremely difficult to describe and interpret. The mathematical technique of asymptotic ray theory is developed in this chapter. Although the details appear complicated, e.g. the higher-order terms in dynamic ray theory, it must be remembered that usually only the lowest-order terms are needed and used. These are relatively easy to understand and compute. Ray theory in a continuous medium consists of three parts which are developed in this chapter for acoustic, isotropic and anisotropic elastic media: kinematic ray theory that describes the geometry and times of rays and wavefronts; dynamic ray theory that describes the geometrical spreading of rays and the displacement magnitude; and polarization theory describing the displacement direction. The chapter concludes with a pretty example of ray tracing in an anisotropic medium with a linear gradient, which illustrates that even in this simple example, complicated non-intuitive results occur.