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 δ(H,n) be the minimum δ such that any k-graph G on n vertices with minimum codegree δ(G)≥δ contains a perfect H -packing. The problem of determining δ(H,n) has been widely studied for graphs (i.e. 2-graphs), but little is known for k≥3. Here we determine the asymptotic value of δ(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 Rödl and Ruciński on the value of δ(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.