Abstract. Biaxial nematic (N b ) liquid crystals are a fascinating condensed matter phase that has baffled for more than thirty years scientists engaged in the challenge of demonstrating its actual existence, and which has only recently been experimentally found. During this period computer simulations of model N b have played an important role, both in providing the basic physical properties to be expected from these systems, and in giving clues about the molecular features essential for the thermodynamic stability of N b phases. However, simulations studies are expected to be even more crucial in the future for unravelling the structural features of biaxial mesogens at molecular level, and for helping in the design and optimisation of devices towards the technological deployment of N b materials. This review article gives an overview of the simulation work performed so far, and relying on the recent experimental findings, focusses on the still unanswered questions which will determine the future challenges in the field.