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SUMMARY This paper is in homage to Robert Stoneley for his many contributions to the analysis of acoustic wave propagation and, in particular, for his introduction of anisotropy into the computation of elastic surface-wave characteristics. The latter has proved to be fundamental to the industry which has grown up around the signal processing devices using surface acoustic waves. The problem reviewed here is that of the reflection of incident acoustic waves from planar interfaces of quite general geometries. The interface may consist of substrate and superstrate welded together, of periodic or non-periodic intervening layers between superstrate and substrate, of free surfaces or of layered free surfaces. Surface waves and layer waves are also included. The materials involved may be of arbitrary crystalline symmetry and may be piezoelectric. The goal is to summarize this range of reflection and transmission problems in a common transfer matrix formalism which is convenient for programming. The same format is used for all the problems and the size of the matrices depends not on the complexity of the interface geometry but only on the crystalline symmetry. The reflection results are expressed as scattering matrices linking the amplitudes of the modes propagating in the substrate and the superstrate.