Abstract: Let $G$ and $H$ be graphs or hypergraphs. A perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. In the simplest case, where $H$ is the graph consisting of a single edge, a perfect $H$-packing in $G$ is simply a perfect matching in $G$; Dirac's theorem tells us that such a packing must exist if $G$ has minimum degree at least $n/2$ (where $n$ is the number of vertices of $G$). The problem of what minimum degree is needed to ensure a perfect $H$-packing in $G$ for general graphs $H$ was then tackled by many researchers, before K\"uhn and Osthus finally established the correct threshold for all graphs $H$ (up to an additive constant). However, for $k$-uniform hypergraphs (or $k$-graphs) much less is known. The case of a perfect matching has been well-studied, but apart from this there were previously no known asymptotically correct results on the minimum degree needed to ensure a perfect $H$-packing in $G$ for $k > 4$ (for any of the various common generalisations of the notion of degree to the $k$-graph setting). In this talk I will demonstrate, for any complete $k$-partite $k$-graph $H$, the asymptotically best-possible minimum codegree condition for a $k$-graph $G$ which ensures that $G$ contains a perfect $H$-packing. This condition depends on the sizes of the vertex classes of $H$, and whether these sizes, or their differences, share any common factors greater than one.