Nematic liquid crystals (NLCs) are anisotropic material intermediate that combine the fluidity of liquids with the orientational order of solids. NLCs are best known for their applications in the thriving liquid crystal display industry. They have tremendous potential in nanoscience, biophysics and material design, all of which rely on scientific computing, applied analysis and mathematical modelling for studying stable/unstable NLC states, switching mechanisms, and dynamical processes on energy landscapes. Within the Landau-de Gennes [1] framework, we study the effects of geometry, anisotropic elasticity and magnetic field on the stable states. We systematically investigate the solution landscapes for NLCs, through which we report new stable states, high-index unstable states with multiple interior defects and novel transition pathways. In addition, we work on modelling complex biological LC systems e.g. membrane fusion and the interaction of nematic tactoids with bacteria [2].