An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs H and G, the anti-Ramsey number ar(G, H) of H in G is the maximum number of colors needed to color the edges of G so that there does not exist a rainbow copy of 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 (K n1,...,nr , M k ) of k-matchings in complete r-partite r-uniform hypergraphs and show the uniqueness of the extremal coloring. Also, we show that K k−1,n2,...,nr is the unique extremal hypergraph for Turán number ex r (K n1,...,nr , M k ) and show that ar r (K n1,...,nr , M k ) = ex r (K n1,...,nr , M k−1 ) + 1, which gives a multi-partite version result of Özkahya and Young's conjecture.