I will discuss inverse problems on a closed Riemannian surface whose geodesic flow is Anosov. More specifically, I will try to establish spectral rigidity (a problem that has been open for some time) and surjectivity results for the adjoint of the geodesic ray transform. These surjectivity results imply the existence of many geometric distributions in H^{-1} invariant under the geodesic flow and play a key role to prove spectral rigidity. This is joint work with Mikko Salo and Gunther Uhlmann.