The set of straight lines of projective three-space is a four-dimensional manifold with a geometric structure induced by the underlying geometry of the projective space P 3. This projective geometry of lines is easily understood in the Klein model, where the lines of P 3 are identified with the points of a certain quadric in the projective 5-space P 5. This Klein quadric is a special case of a Grassmann manifold, a concept which is studied in Sec.2.2. The lines of Euclidean space with their metric properties lead to a different model, the Study sphere. Here oriented lines of E 3 are identified with the points of a unit sphere constructed with dual numbers instead of real ones. Therese models are closely tied to the problem of introducing suitable coordinates for lines: The Klein quadric corresponds to Plücker coordinates of lines, which are a special case of Grassmann coordinates of subspaces. The Study sphere leads to dual coordinate vectors for oriented lines. Working in a geometric point model enables better understanding and a simple interpretation of various objects of line space. The design of efficient algorithms involving lines is greatly simplified if it is based on the right geometric model.