Nematic liquid crystals are classical examples of partially ordered materials that combine fluidity with the order of crystalline solids. They are the working material of a range of electro-optic devices i.e. in the liquid crystal display industry and more recently, they are used in sensors, actuators, elastomers, security applications and pathological studies. We review the celebrated Landau-de Gennes theory for nematic liquid crystals and focus on the modelling of nematics confined to thin quasi-2D systems, with reference to 2D polygons. We perform asymptotic analysis in certain distinguished limits, encoded in terms of geometrical, material and temperature-dependent parameters, accompanied by exhaustive studies of solution landscapes that include stable and unstable solution branches for these systems. In the last leg of the talk, we discuss the mathematical modelling of some recent experiments on nematic shells and doped bent-core liquid crystals, to illustrate the synergistic links between theory, experiment and novel applications. All collaborations will be acknowledged throughout the talk.