  This paper establishes dynamical localization properties of certain families
of unitary random operators on the d-dimensional lattice in various regimes.
These operators are generalizations of one-dimensional physical models of
quantum transport and draw their name from the analogy with the discrete
Anderson model of solid state physics. They consist in a product of a
deterministic unitary operator and a random unitary operator. The deterministic
operator has a band structure, is absolutely continuous and plays the role of
the discrete Laplacian. The random operator is diagonal with elements given by
i.i.d. random phases distributed according to some absolutely continuous
measure and plays the role of the random potential. In dimension one, these
operators belong to the family of CMV-matrices in the theory of orthogonal
polynomials on the unit circle.
  We implement the method of Aizenman-Molchanov to prove exponential decay of
the fractional moments of the Green function for the unitary Anderson model in
the following three regimes: In any dimension, throughout the spectrum at large
disorder and near the band edges at arbitrary disorder and, in dimension one,
throughout the spectrum at arbitrary disorder. We also prove that exponential
decay of fractional moments of the Green function implies dynamical
localization, which in turn implies spectral localization.
  These results complete the analogy with the self-adjoint case where dynamical
localization is known to be true in the same three regimes.
