Mesogenic molecules invariably deviate from the cylindrical symmetry usually assumed for them. One consequence of this is that, in addition to the ubiquitous uniaxial nematic phase, there should also be a biaxial nematic phase. For this the re are three different principal components. of a second rank tensorial property, such as the magnetic susceptibility, as opposed to two for a uniaxial phase. Each of these components is associated with a separate director; the three directors correspond to the directions about which three orthogonal axes set in the molecule tend to be aligned. Although the existence of the biaxial nematic phase was predicted over 30 years ago it was not until 10 years later that the first claim to have found this phase appeared. This system was, in fact, a lyotropic liquid crystal. Surprisingly, the search for a thermotropic biaxial nematic has proved to be especially challenging. This review is concerned with the thermotropic materials for which a biaxial nematic have now been claimed. Of particular importance is the technique used to establish the symmetry of the nematic phase and it is suggested that deuterium NMR spectroscopy is a powerful method with which to determine this. However, the biaxial nematics claimed to be formed by certain compounds are shown by NMR to have uniaxial symmetry. The reasons why the biaxial nematic phase proves to be so elusive are explored using molecular field theory and used to inform the design of thermotropic mesogens which are likely to form this phase.