  Let $G$ and $H$ be $k$-graphs ($k$-uniform hypergraphs); then a perfect
$H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$
which together cover every vertex of $G$. For any fixed $H$ let $\delta(H, n)$
be the minimum $\delta$ such that any $k$-graph $G$ on $n$ vertices with
minimum codegree $\delta(G) \geq \delta$ contains a perfect $H$-packing. The
problem of determining $\delta(H, n)$ has been widely studied for graphs (i.e.
$2$-graphs), but little is known for $k \geq 3$. Here we determine the
asymptotic value of $\delta(H, n)$ for all complete $k$-partite $k$-graphs $H$,
as well as a wide class of other $k$-partite $k$-graphs. In particular, these
results provide an asymptotic solution to a question of R\"odl and Ruci\'nski
on the value of $\delta(H, n)$ when $H$ is a loose cycle. We also determine
asymptotically the codegree threshold needed to guarantee an $H$-packing
covering all but a constant number of vertices of $G$ for any complete
$k$-partite $k$-graph $H$.

Combinatorics