  We prove that for fixed $k$, every $k$-uniform hypergraph on $n$ vertices and
of minimum codegree at least $n/2+o(n)$ contains every spanning tight $k$-tree
of bounded vertex degree as a sub\-graph. This generalises a well-known result
of Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. Our result is
asymptotically sharp. We also prove an extension of our result to hypergraphs
that satisfy some weak quasirandomness conditions.
