Bernd Sturmfels, a mathematician at UC Berkeley, gives this talk at the Institute for Advanced Study as part of the MPG-IAS-NTU Center kick-off conference, taking the classical geometry of lines in three-dimensional space as his starting point. He works through how a line in 3-space is encoded by Plucker coordinates and how the set of all such lines forms the Grassmannian, a curved surface sitting inside projective space. From there he traces the algebraic relations that describe configurations of several lines at once, the kind of structure that shows up in robotics, computer vision, and optimization as much as in pure algebraic geometry. Sturmfels moves between blackboard computation and concrete small examples, the style he is known for, building toward why a subject this old keeps generating new research. The talk runs just under fifty minutes and assumes some background in linear and projective algebra.