To obtain results that are better than ray theory and remain valid at singularities, solutions of the full wave equations are needed. In a one-dimensional or stratified medium, there is an exact procedure to obtain these - transformation of the wave equation to reduce the partial differential equation to an ordinary differential equation; solution of this using one of several well-developed techniques; and inversion of the results from the transform domain to obtain the response. In this chapter, we develop the ordinary differential systems for acoustic, isotropic and anisotropic, elastic media. The important ray expansion is then introduced, to expand this into propagators for each continuous layer of the model. Three techniques that can be used to solve the ordinary differential equations when the layers are heterogeneous, are described: the WKBJ asymptotic expansion, the WKBJ iterative solution or Bremmer series, and the Langer asymptotic expansion. These methods are useful to describe various canonical solutions. However, for realistic media, a combination of methods might well be required and it is often more realistic to resort to numerical methods to solve the ordinary differential equations.