This chapter discusses the theory and applications of complex rays. Complex rays are solutions of the ray equations of traditional geometrical optics but correspond to extremals in the six-dimensional complex space. The chapter discusses the basic equations for ordinary and complex geometrical optics, the properties of complex rays, and the selection rules associated with them. These results follow from the application of standard asymptotic methods, such as stationary phase and saddle-point methods, to the Kirchhoff solutions for wave propagation. The chapter presents examples of complex trajectories in different optical and physical problems and a complex ray analysis of Gaussian beam propagation. The chapter also discusses certain distinctive aspects of complex geometrical optics, including nonlocality and applicability. These considerations serve to give some measure of the physical significance of a complex ray.