In his classic paper on the elastic behaviour of nematic liquid crystals published in 1958, Sir Charles Frank hints at the existence of a nematic phase with biaxial symmetry in addition to the conventional nematic phase with its uniaxial symmetry. Little notice appears to have been taken of this intriguing suggestion until 1970 when Freiser presented a molecular theory of nematics formed from biaxial molecules. This predicted that such materials should exhibit isotropic, uniaxial nematic and biaxial nematic phases although for a certain critical biaxiality the system should pass directly from the isotropic phase to a biaxial nematic. Despite these clear predictions it was not until the late 1980s that the claims to have prepared compounds exhibiting a thermotropic biaxial nematic phase began to appear; these originated from the laboratories of Jaques Malthtte, S. Chandrasekhar and Klaus Praefcke. However, subsequent studies of some of the materials suggested that the nematic phases claimed to be biaxial were, in fact, uniaxial. Although the formation of real biaxial nematics is in doubt, computer simulators have produced biaxial nematic phases in keeping with the theoretical predictions. In addition, about 17 years ago Alfred Saupe had discovered a biaxial nematic phase formed by a lyotropic system.