Given a Riemannian manifold, the geodesic X-ray transform of a symmetric tensor field is defined by the line integrals of the latter over geodesics, and it is a central object in geometric inverse problems. Broadly speaking, one is interested in recovering information about the unknown tensor field from its X-ray transform, and the extent to which this is possible depends heavily on the underlying geometry. In this talk, we focus on the geodesic X-ray transform in the geometric setting of two-dimensional asymptotically hyperbolic manifolds, which are non-constant curvature generalizations of hyperbolic space. This setting is interesting in part due to its connections to theoretical physics, specifically the AdS-CFT Correspondence. The geodesic X-ray transform on symmetric tensor fields of positive rank has a natural nullspace, implying that such a tensor field cannot be uniquely recovered from its X-ray transform. On asymptotically hyperbolic surfaces we propose gauge representatives modulo the nullspace of the transform to be reconstructed from the data, by proving a ``transverse traceless-conformal-potential'' decomposition. We then use our tensor decompositions to provide range characterizations of the geodesic X-ray transform in the special case of 2-dimensional hyperbolic space, as well as to develop reconstruction procedures. Based on joint work with François Monard and Yuzhou Zou.