A closed Riemannian manifold is said to be Anosov if its geodesic flow on its unit tangent bundle is Anosov (also called uniformly hyperbolic in the literature). Typical examples are provided by negatively-curved manifolds. On such manifolds, the X-ray transform is simply defined as the integration of continuous functions along periodic geodesics. I will review some recent results on the analytic study of the X-ray transform (in particular, stability estimates). The techniques rely on microlocal tools introduced by Guillarmou and further investigated by Guillarmou-Lefeuvre, and on new finite and approximate Livsic theorems proved by Gouëzel-Lefeuvre. If time permits, I will explain how these results can be applied to prove the local rigidity of the marked length spectrum.