A central problem in extremal graph theory is to study degree conditions that force a graph G to contain a copy of some large or even spanning graph F. One of the most classical results in this area is Dirac's theorem on Hamilton cycles. An extension of this theorem is the Posa-Seymour conjecture on powers of Hamilton cycles, which has been proved for large graphs by Komlos, Sarkozy and Szemeredi. Extension of these results to hypergraphs, using codegree conditions and tight (powers of) cycles, have been studied by various authors. We give an overview of the known results, and then show a codegree condition which is sufficient for ensuring arbitrary powers of tight Hamilton cycles, for any uniformity. This could be seen as an approximate hypergraph version of the Posa-Seymour conjecture. On the way to our result, we show that the same codegree conditions are sufficient for finding a copy of every spanning hypergraph of bounded tree-width which admits a tree decomposition where every vertex is in a bounded number of bags. This is joint work with Nicolas Sanhueza-Matamala and Matias Pavez-Signe.