  Let $H$ be a $k$-partite $k$-graph with $n$ vertices in each partition class,
and let $\delta_{k-1}(H)$ denote the minimum co-degree of $H$. We characterize
those $H$ with $\delta_{k-1}(H) \geq n/2$ and with no perfect matching. As a
consequence we give an affirmative answer to the following question of R\"odl
and Ruci\'nski: If $k$ is even or $n \not\equiv 2 \pmod 4$, does
$\delta_{k-1}(H) \geq n/2$ imply that $H$ has a perfect matching? We also give
an example indicating that it is not sufficient to impose this degree bound on
only two types of $(k-1)$-sets.
