The mathematical analysis of liquid crystal models is difficult, as can be seen by their close relationship to the study of singularities for harmonic maps. In this talk, I will present several mathematical models used to study liquid crystals, as well as the connection with classical results on harmonic maps. I will then present new results on energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle in the context of the Landau de-Gennes energy. The Landau de-Gennes model allows for a greater variety of singularities than is allowed in the Oseen-Frank model, which is related to harmonic maps with values in the unit sphere.