  The extremal problems regarding the maximum possible size of intersecting
families of various combinatorial objects have been extensively studied. In
this paper, we investigate supersaturation extensions, which in this context
ask for the minimum number of disjoint pairs that must appear in families
larger than the extremal threshold. We study the minimum number of disjoint
pairs in families of permutations and in $k$-uniform set families, and
determine the structure of the optimal families. Our main tool is a removal
lemma for disjoint pairs. We also determine the typical structure of
$k$-uniform set families without matchings of size $s$ when $n \ge 2sk +
38s^4$, and show that almost all $k$-uniform intersecting families on vertex
set $[n]$ are trivial when $n\ge (2+o(1))k$.
