The research described in this thesis covers various aspects of the forward calculation of seismic ray fields and ray field maps. The central theme is the solution of problems encountered in smooth but complex media, i.e., media that give rise to wave front folding and associated multi-pathing of rays. The ultimate aim of the presented material is to enhance the efficiency of seismic inverse methods, by enhancing the efficiency of the forward calculations. Particular emphasis is placed on the applicability of the ray tracing results to seismic inverse methods. After an overview of seismic ray theory in Chapter 2, a novel approach to the calculation and representation of ray field maps is introduced in Chapter 3. The approach is particularly useful in cases where ray field maps are needed for a dense distribution of sources at an acquisition surface, as in reflection seismics and borehole tomography. For such source distributions it is suggested to construct a single ray field map in an extended space of spatial coordinates and angles, rather than a number of maps in the spatial domain for a range of acquisition coordinates. The ray field map in the position/angle domain is single-valued, regardless of the complexity of the medium and the ray field information is organised by angles at depth rather than by points of emergence at the surface, which makes the maps particularly suitable for use in modern seismic imaging methods. An important result is that, in contrast to what is commonly assumed, obtaining this information does not require the tracing of rays up towards the acquisition surface. Instead, existing algorithms that trace downwards can be adapted to work in the position/angle domain, leading to a considerable gain in efficiency. Interpolation is an important tool in both the construction and the application of ray field maps. A new technique for accurate interpolation using derivative information is presented in Chapter 4. It is a hybrid of extrapolation to arbitrary order and linear interpolation, and ...