  In this article we shall derive functional limit theorems for the
multi-dimensional elephant random walk (MERW) and thus extend the results
provided for the one-dimensional marginal by Bercu and Laulin (2019). The MERW
is a non-Markovian discrete time-random walk on $\mathbb{Z}^d$ which has a
complete memory of its whole past, in allusion to the traditional saying that
an elephant never forgets. As the name suggests, the MERW is a $d$-dimensional
generalisation of the elephant random walk (ERW), the latter was first
introduced by Sch\"utz and Trimper in 2004. We measure the influence of the
elephant's memory by a so-called memory parameter $p$ between zero and one. A
striking feature that has been observed by Sch\"utz and Trimper is that the
long-time behaviour of the ERW exhibits a phase transition at some critical
memory parameter $p_c$. We investigate the asymptotic behaviour of the MERW in
all memory regimes by exploiting a connection between the MERW and P\'olya
urns, following similar ideas as in the work by Baur and Bertoin for the ERW.
