Abstract The global analog for a manifold of a linear first order partial differential operator with complex coefficients is a complex line sub-bundle of the complexified tangent bundle. I will discuss which bundles arise in this way and how the characteristic points of the operator influence the structure of the line bundle. I will review the Chern class of a line bundle and provide a new interpretation of this class for bundles on compact manifolds of dimension two. There are many open problems, mainly topological in nature, for systems of operators.