  An edge-colored hypergraph is rainbow if all of its edges have different
colors. Given two hypergraphs $\mathcal{H}$ and $\mathcal{G}$, the anti-Ramsey
number $ar(\mathcal{G}, \mathcal{H})$ of $\mathcal{H}$ in $\mathcal{G}$ is the
maximum number of colors needed to color the edges of $\mathcal{G}$ so that
there does not exist a rainbow copy of $\mathcal{H}$. Li et al. determined the
anti-Ramsey number of $k$-matchings in complete bipartite graphs. Jin and Zang
showed the uniqueness of the extremal coloring. In this paper, as a
generalization of these results, we determine the anti-Ramsey number
$ar_r(\mathcal{K}_{n_1,\ldots,n_r},M_k)$ of $k$-matchings in complete
$r$-partite $r$-uniform hypergraphs and show the uniqueness of the extremal
coloring. Also, we show that $\mathcal{K}_{k-1,n_2,\ldots,n_r}$ is the unique
extremal hypergraph for Tur\'{a}n number
$ex_r(\mathcal{K}_{n_1,\ldots,n_r},M_k)$ and show that
$ar_r(\mathcal{K}_{n_1,\ldots,n_r},$
$M_k)=ex_r(\mathcal{K}_{n_1,\ldots,n_r},M_{k-1})+1$, which gives a
multi-partite version result of \"Ozkahya and Young's conjecture.
