Liquid crystals are phases of matter intermediate between typical liquids and crystalline solids, in terms of their molecular ordering and physical properties. In recent years, there has been a growing realization that liquid crystalline order emerges not just in equilibrium materials but also in the collective motions of many crowded, interacting objects far from thermal equilibrium—including in bacterial colonies, cellular tissues, and biofilament suspensions. In such “active matter” systems, the generic dynamics are determined to a great extent by the topological defects of liquid crystals, which in two dimensions are analogous to vortices in magnetic and superfluid systems. In this talk I will describe how this coupling of dynamics to topology manifests in three-dimensional active matter, and I will explore how these findings pose new questions for the mathematics and physics of liquid crystals.