Physical systems with topological defects offer a sandbox for purely mathematical ideas in geometry and topology; in liquid crystalline phases of matter, the underlying topology can be literally seen!  In this talk, I will review the standard classification of topological defects according to homotopy theory.  This description will be applied specifically to defects in nematic liquid crystals, and further topological characterization of ring and knot defects will be presented.  I will also demonstrate why the homotopy-theoretic approach fails to describe systems with broken translational symmetry and the alternative tools from topology required to understand their defects.