This paper looks at the algorithmic complexity of two problems on hypergraphs. The first problem is the problem to test whether a given hypergraph is saturated, i.e., whether every subset of the vertex set is contained in an edge or contains an edge of the hypergraph. The second problem is the problem to test whether a given hypergraph is the transversal hypergraph of another given hypergraph. A transversal of a hypergraph \(H\) is a subset of the vertices that intersects each edge of \(H\). It is minimal if it does not contain another transversal as a proper subset. The collection of all minimal transversals of \(H\) forms the transversal hypergraph of \(H\). It is shown that the hypergraph saturation problem is co-NP-complete, even for hypergraphs that are self-complemented (the complement of itself), and all edges have size \(k\) or \(n- k\), for fixed \(k\geq 3\). However, for self-complemented hypergraphs with all edges of size 2 or \(n- 2\), the problem is shown to be solvable in polynomial time. Also, the paper looks at the hypergraph saturation problem for simple hypergraphs, i.e., no edge can be a proper subset of another edge. It is shown that this problem is as hard as the subcase for self-complemented hypergraphs. As a byproduct, several hardness results are obtained for the problem to color the vertices of a hypergraph with two colors, such that each edge contains vertices of both colors. The following results are shown for the transversal-hypergraph problem. First, if this problem is not solvable in polynomial time, then there exists no algorithm for computing the transversal hypergraph that uses time, polynomial in the size of the output. Several characterizations of the transversal-hypergraph problem are obtained, also in terms of saturation. Then, it follows that there is a polynomial time transformation from hypergraph saturation for simple graphs to the transversal-hypergraph problem. Also, it is shown that the problem to decide whether a given hypergraph is its own transversal is equivalent to the hypergraph saturation problem for simple hypergraphs. Finally, several special cases are established that allow polynomial time algorithms: if the sizes of edges differ by at most a constant, then the hypergraph saturation problem is polynomial time solvable; if the sizes of the edges of one of the input graphs of the transversal-hypergraph problem is bounded by a constant, then this problem allows a polynomial time algorithm; and a polynomial time algorithm is given for computing the transversal of acyclic hypergraphs, implying the polynomial time solvability of the transversal-hypergraph problem for acyclic hypergraphs. Several applications, for satisfiability problems, problems in relational databases, distributed databases, Boolean switching theory, and model-based diagnosis are given.