  We study the random walk $X$ on the range of a simple random walk on
$\mathbb{Z}^d$ in dimensions $d\geq 4$. When $d\geq 5$ we establish quenched
and annealed scaling limits for the process $X$, which show that the
intersections of the original simple random walk path are essentially
unimportant. For $d=4$ our results are less precise, but we are able to show
that any scaling limit for $X$ will require logarithmic corrections to the
polynomial scaling factors seen in higher dimensions. Furthermore, we
demonstrate that when $d=4$ similar logarithmic corrections are necessary in
describing the asymptotic behaviour of the return probability of $X$ to the
origin.
