  In 1964, Erd\H{o}s, Hajnal and Moon introduced a saturation version of
Tur\'an'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 Erd\H{o}s-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 Erd\H{o}s-Szekeres theorem on convex $k$-gons, as
well as multiple semisaturation theorems for sequences and posets.
