We develop a notion of axis in the Culler–Vogtmann outer space Xr of a finite rank free group Fr, with respect to the action of a fully irreducible outer automorphism phi. Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmuller space, Xr has no natural metric, and phi seems not to have a single natural axis. Instead our axes for phi, while not unique, fit into an “axis bundle” Aφ with nice topological properties: Aφ is a closed subset of Xr proper homotopy equivalent to a line, it is invariant under phi, the two ends of Aφ limit on the repeller and attractor of the source–sink action of phi on compactified outer space, and Aφ depends naturally on the repeller and attractor. We propose various definitions for Aφ, each motivated in different ways by train track theory or by properties of axes in Teichmuller space, and we prove their equivalence.