It has become increasingly clear in recent years that the two materials most commonly used in surface acoustic wave (SAW) devices (α-quartz and lithium niobate) have serious shortcomings for certain applications. For example, the relatively poor electromechanical coupling of α-quartz renders it unsuitable for devices requiring large bandwidths while the poor temperature coefficients of SAW velocities on LiNbO3 renders this material unsuitable for many applications where environmental stability is important. Another important device limitation, which is determined by the properties of these materials, is the upper frequency limit of operation. This is currently fixed at about 1 GHz by the SAW velocities on LiNbO3 and α-quartz of ~3400 and ~3200 m s-1 respectively. A large amount of work in recent years has gone into investigating the piezoelectric and elastic properties of new materials, with a view to finding one which either combines high electromechanical coupling with a zero temperature coefficient of SAW velocity or shows a significantly higher SAW velocity than α-quartz or LiNbO3. The results of some of this work are reviewed and discussed in terms of the materials' crystal structures and the problems presented by their crystal growth. The prospects for new SAW and other piezoelectric devices fabricated using these materials are also discussed.