A line field on a manifold is a smooth and continuous assignment of a tangent line to each point on the manifold. The projective span of a smooth manifold is the maximal number of linearly independent line fields. After motivating the study of this fascinating numerical invariant, I will not only give a necessary condition for the existence of linearly independent line fields on (almost) complex manifolds, but I will also explain that this condition is additionally sufficient in certain cases. Finally, we will apply these necessary and sufficient conditions to obtain a refinement of the Schwarzenberger condition that dictates which cohomology classes can be the Chern classes of a complex vector bundle (with prescribed line bundle splitting properties) over complex projective space. This is joint work with Nikola Sadovek (Dresden) based on [arXiv:2411.14161] (recently accepted for publication in Mathematische Annalen).