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We prove a conjecture by Bertoin that the multi-dimensional elephant randomwalk on $\mathbb{Z}^d$($d\geq 3$) is transient and the expected number of zerosis finite. We also provide some estimates on the rate of escape. In dimensions$d= 1, 2$, we prove that phase transitions between recurrence and transienceoccur at $p=(2d+1)/(4d)$.  Let $S$ be an elephant random walk with parameter $p$. For $p \leq 3/4$, weprovide a Berry-Esseen type bound for properly normalized $S_n$. For $p>3/4$,the distribution of $\lim_{n\to \infty} S_n/n^{2p-1}$ will be studied.

Recurrence and transience of multidimensional elephant random walks