  The dynamics of a one dimensional quantum walker on the lattice with two
internal degrees of freedom, the coin states, is considered. The discrete time
unitary dynamics is determined by the repeated action of a coin operator in
U(2) on the internal degrees of freedom followed by a one step shift to the
right or left, conditioned on the state of the coin. For a fixed coin operator,
the dynamics is known to be ballistic. We prove that when the coin operator
depends on the position of the walker and is given by a certain i.i.d. random
process, the phenomenon of Anderson localization takes place in its dynamical
form. When the coin operator depends on the time variable only and is
determined by an i.i.d. random process, the averaged motion is known to be
diffusive and we compute the diffusion constants for all moments of the
position.
