  In the randomly-oriented Manhattan lattice, every line in $\mathbb{Z}^d$ is
assigned a uniform random direction. We consider the directed graph whose
vertex set is $\mathbb{Z}^d$ and whose edges connect nearest neighbours, but
only in the direction fixed by the line orientations. Random walk on this
directed graph chooses uniformly from the $d$ legal neighbours at each step. We
prove that this walk is superdiffusive in two and three dimensions. The model
is diffusive in four and more dimensions.
