  We study the elephant random walk in arbitrary dimension $d\geq 1$. Our main
focus is the limiting random variable appearing in the superdiffusive regime.
Building on a link between the elephant random walk and P\'olya-type urn
models, we prove a fixed-point equation (or system in dimension two and larger)
for the limiting variable. Based on this, we deduce several properties of the
limit distribution, such as the existence of a density with support on $\mathbb
R^d$ for $d\in\{1,2,3\}$, and we bring evidence for a similar result for $d\geq
4$. We also investigate the moment-generating function of the limit and give,
in dimension $1$, a non-linear recurrence relation for the moments.
