Liquid crystal materials are of great importance in the display industry because of their optical properties and the ease with which their orientational properties can be altered by externally applied electric or magnetic fields. The determination and prediction of the orientational properties of liquid crystals in microscopic confinements is quite challenging. This is due to the complexity of the models; the competing influences of intermolecular forces, external fields, and boundary interactions; the inherent nonlinearity of the problem and accompanying structural phase transitions; and the presence of defects and singularities. We will survey the main continuum and phenomenological models for liquid crystals and will discuss recent numerical results concerning the fine structure of point defects in spherical droplets of nematic liquid crystals. For these numerics, we will use a tensor-order-parameter model, for which the equilibrium conditions lead to a coupled system of semilinear elliptic partial differential equations. We solve a symmetry-reduced system using a finite-element package. Phase diagrams and bifurcation diagrams illustrating the relationships among three competing solutions are obtained.