Rational curves are one of the main tools in symplectic geometry and provide a bridge to algebraic geometry. Complex lines are a more general class of curve that has the potential to connect symplectic and complex analytic geometry. These curves are non-compact, which presents a serious difficulty in understanding their symplectic aspects. In this talk, I will explain how Ahlfors currents can be used to resolve this difficulty and produce a theory parallel to that of rational curves. In particular, Ahlfors currents can be constructed via a continuity method, they control bubbling of holomorphic curves, and they form a convex set.