I will focus on recent results on behavior of minimizers of some common models for liquid crystals stemming from the seminal work of Brezis, Bethuel and Helein on the Ginzburg-Landau theory. In particular, I will describe a recent result for Ginzburg-Landau theory posed on a singular surface (a cone) to explore the role of a singularity on a surface in determining vortex location. I will also discuss a model for nematic/isotropic phase transitions in which there emerge both line defects and singular phase boundaries.