The celebrated Livšic theorem [A. N. Livšic, Certain properties of the homology of Y-systems, Mat. Zametki 10 (1971), 555–564; A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320] states that given a manifold M, a Lie group G, a transitive Anosov diffeomorphism f on M and a Hölder function η:M↦G whose range is sufficiently close to the identity, it is sufficient for the existence of ϕ:M↦G satisfying η(x)=ϕ(f(x))ϕ(x)−1 that a condition—obviously necessary—on the cocycle generated by η restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.