Complex lines are a class of pseudoholomorphic curves which generalize rational curves. Applications of complex lines to symplectic geometry have been proposed, but they remain poorly understood. In this talk, I will describe a framework for studying them based on Ahlfors currents. These currents allow a topological study of complex lines which is largely analogous to that of rational curves. As an application, I will discuss a construction of complex lines under only topological assumptions, generalizing a theorem of Bangert.