Summary: ``We say that a tree $T$ is an $S$-Steiner tree if $S\subseteq V(T)$ and a hypergraph is an $S$-Steiner hypertree if it can be trimmed to an $S$-Steiner tree. We prove that it is NP-complete to decide, given a hypergraph $\Cal H$ and some $S\subseteq V(\Cal H)$, whether there is a subhypergraph of $\Cal H$ which is an $S$-Steiner hypertree. As corollaries, we give two negative results for two Steiner orientation problems in hypergraphs. Firstly, we show that it is NP-complete to decide, given a hypergraph $\Cal H$, some $r\in V(\Cal H)$ and some $S\subseteq V(\Cal H)$, whether this hypergraph has an orientation in which every vertex of $S$ is reachable from $r$. Secondly, we show that it is NP-complete to decide, given a hypergraph $\Cal H$ and some $S\subseteq V(\Cal H)$, whether this hypergraph has an orientation in which any two vertices in $S$ are mutually reachable from each other. This answers a longstanding open question of the Egerváry Research group. We further show that it is NP-complete to decide if a given hypergraph has a well-balanced orientation. On the positive side, we show that the problem of finding a Steiner hypertree and the first orientation problem can be solved in polynomial time if the number of terminals $|S|$ is fixed.''