  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 $n\equiv 0\pmod k$. 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.
