Asymptotically hyperbolic manifolds are a class of non-compact, complete manifolds whose behavior near infinity resembles in many ways that of hyperbolic space. In part due to the behavior of geodesics and their infinite length, and the fact that those spaces have infinite volume, the study of the geodesic X-ray transform becomes particularly interesting and challenging. In this talk we will discuss the problem of injectivity of the geodesic X-ray transform on asymptotically hyperbolic manifolds, identify the issues induced by the geometry, and present some progress in extending to this setting the celebrated local injectivity result proved by Uhlmann and Vasy on compact manifolds with boundary. Joint work with Robin Graham.