The propagator for coupled-mode elastic waves can be cast into a number of different representations, which emphasize particular aspects of the wave propagation in a laterally heterogeneous medium. One representation has the form of a generalized scattering operator and contains a quantity that can be interpreted as the lateral impedance. Another representation reduces naturally to the JWKB approximation for smoothly varying media with no mode coupling. The propagator solution for the fields in a laterally heterogeneous elastic medium with weak random boundary fluctuations leads naturally to the application of Feynman diagram techniques for the derivation of Dyson's equation and the Bethe–Salpeter equation for the propagator mean and covariance, respectively. The diagram techniques are reviewed and their utility for solution of random media elastic wave problems is demonstrated.