Keywords: Nematic Liquid Crystal, non-Newtonian, lubrication theory, asymptotics, electric field effects, interfacial instability, free boundary problem, LCD Abstract: Nematic liquid crystals are materials intermediate between the liquid and solid states. They are typically composed of long, rod-like molecules, which have a tendency to align with their neighbors, imparting a short-range (but no long-range) orientational order - although nematics flow, like conventional liquids, they also retain some elastic character. This gives rise to complex behavior that does not arise in Newtonian liquids. Moreover, their response to an applied electric field gives nematics wide application in the electronic display industry. We consider mathematical models for three different problems. Two are classical fluid-dynamical setups: a small droplet spreading on a flat substrate; and a two-dimensional "liquid bridge" (or liquid sheet) under tension. These are free boundary problems, and the thin geometry in each case enables the use of "lubrication" analysis to systematically derive reduced mathematical models governing the free surface evolution. The spreading drop analysis leads to variants of the classical 4th order "thin film" equation, which can exhibit instability in certain regimes. The "liquid bridge" problem leads to new versions of the so-called Trouton model for Newtonian viscous sheets. The third problem arises in the display industry, and is concerned with manufacturing a "bistable" display device, that can exhibit two optically-distinct configurations in the absence of an electric field. Such a device has the potential for considerably reducing the power demands of a display, with accompanying benefits for battery lifetime and device portability.