In this thoughtful and stimulating paper the author develops a “nonstationary” normal form theory generalizing the related theorems of G. D. Birkhoff, J. Moser and S. Sternberg. To generalize the hyperbolicity condition to the nonstationary case, Anosov diffeomorphisms are considered, which are hyperbolic at each point. Preservation of area is still required (as in Moser’s case) and plays an important role in proving convergence. Uniqueness for the nonstationary normal form manifests itself in a cohomological uniqueness theorem. The terms of the nonstationary normal form are used to define cocycles and (local and global) invariants for the dynamical system. The author also studies certain skew product dynamical systems.