In certain asymptotic parameter regimes, propagation, radiation and diffraction processes in a linear field may be described locally in terms of plane waves and schematized in terms of rays. Properties of space-time rays descriptive of time-dependent fields in lossless dispersive, isotropic or anisotropic media, and corresponding features of time-harmonic rays, are reviewed, with emphasis on the relation between rays and the plane wave dispersion surfaces characteristic of the medium. Graphical procedures are given for ray tracing in the presence of weak, continuous inhomogeneities in the medium properties; abrupt spatial or temporal medium changes ; and discontinuities exciting certain types of diffracted rays. These aspects are then related to integral representations for the field and to its asymptotic behavior characterized by saddle points and singularities in the integrands. Attention is given to transition regions wherein simple asymptotic methods fail and the ray description must be modified. The ...