The exceptional wave theory developed in the last two decades is applied to trigonal crystals, widely used in surface acoustic wave devices: quartz, lithium niobate, and lithium tantalate. For quartz all crystallographic orientations where one of bulk acoustic waves satisfies the stress‐free mechanical boundary conditions, have been found and plotted on the stereographic projection of a unit wave normal sphere as ‘‘exceptional wave lines.’’ The possibility of existence and the main features of pure shear, quasishear, and quasilongitudinal exceptional waves are examined in three cut families with Euler angles (0°,θ,0°), (90°,90°,ψ), and (0°,θ,90°). These orientations are analyzed both without and with piezoelectric effect by means of analytical and numerical techniques. The obligatory existence of undamped surface skimming bulk waves or leaky waves is proved for selected cuts of trigonal crystals, such as a well‐known 41°‐rotated Y cut of lithium niobate. The behavior of these waves is compared in three crystals. A simple relation between elastic moduli of trigonal crystals with point symmetry of 32 or 3m has been derived, which allows the existence of a longitudinal type of exceptional waves. Such waves have been discovered in quartz. They are similar to those found earlier in lithium tetraborate and in both crystals they give rise to piezoelectrically coupled high velocity leaky waves with small attenuation. © 1996 American Institute of Physics.