Almost perfect matchings in $k$-partite $k$-graphs

The minimum co-degree threshold for a perfect matching in a $k$-graph with $n$ vertices was determined by R"odl, Ruci'nski and Szemer'edi for the case when $n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}k)$. Recently, Han resolved the remaining cases when $n \not\equiv 0\pmod k$, establishing a conjecture of R"odl, Ruci'nski and Szemer'edi. In this paper, we determine the minimum co-degree threshold for almost perfect matchings in $k$-partite $k$-graphs, answering a question of R"odl and Ruci'nski.