Let [Fscr ] be a family of forbidden k-hypergraphs (k-uniform set systems). An [Fscr ]-saturated hypergraph is a maximal k-uniform set system not containing any member of [Fscr ]. As the main result we prove that, for any finite family [Fscr ], the minimum number of edges of an [Fscr ]-saturated hypergraph is O(nk−1). In particular, this implies a conjecture of Tuza. Some other related results are presented.