We formulate a sub-Riemannian X-ray transform on contact manifolds. Such manifolds are modeled locally (via their Gromov-Hausdorff tangent cones) by 2-step nilpotent Lie groups, with the Heisenberg group exhibiting the simplest nontrivial case. We explicitly describe the Heisenberg ray transform, and an ongoing attempt at its inversion using tools from noncommutative harmonic analysis in the spirit of Helgason that rhyme with the Euclidean case. The primary challenge in this geometry inhibiting standard tools is the ubiquitous presence of conjugate points.