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We say a $s$-uniform $r$-partite hypergraph is complete, if it has a vertex partition $\{V_1,V_2,...,V_r\}$ of $r$ classes and its hyperedge set consists of all the $s$-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete $s$-uniform $r$-partite hypergraph with $k$ vertices in each vertex class by ${\cal T}_{s,r}(k)$. In this paper we prove that if $h,\ r$ and $s$ are positive integers with $2\leq s\leq r\leq h$ then there exists a constant $k=k(h,r,s)$ so that if $H$ is an $s$-uniform hypergraph with $h$ vertices and chromatic number $\chi(H)=r$ then any proper edge coloring of ${\cal T}_{s,r}(k)$ has a rainbow $H$-factor.

Rainbow $H$-Factors of Complete $s$-Uniform $r$-Partite Hypergraphs