Abstract In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán’s classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a K r-free, n-vertex graph with the property that the addition of any further edge yields a copy of K r. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős–Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the family to have the forbidden configuration, but insist that any addition to the family yields a new copy of the forbidden configuration. In this setting, we prove a semisaturation version of the Erdős–Szekeres theorem on convex k-gons, as well as multiple semisaturation theorems for sequences and posets.