  Erd\H{o}s-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of
families of sets, subject to various intersection requirements on the sets in
the family. Stability versions of such theorems assert that if the size of a
family is close to the maximum possible size, then the family itself must be
close (in some appropriate sense) to a maximum-sized family.
  In this paper, we present an approach to obtaining stability versions of
EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube.
Our approach is rather general, and allows the leveraging of a wide variety of
exact EKR-type results into strong stability versions of these results, without
going into the proofs of the original results.
  We use this approach to obtain tight stability versions of the EKR theorem
itself and of the Ahlswede-Khachatrian theorem on $t$-intersecting families of
$k$-element subsets of $\{1,2,\ldots.n\}$ (for $k < \frac{n}{t+1}$), and to
show that, somewhat surprisingly, all these results hold when the intersection
requirement is replaced by a much weaker requirement.
  Other examples include stability versions of Frankl's recent result on the
Erd\H{o}s matching conjecture, the Ellis-Filmus-Friedgut proof of the
Simonovits-S\'{o}s conjecture, and various EKR-type results on $r$-wise
(cross)-$t$-intersecting families.
