  A collection of sets is {\em intersecting} if every two members have nonempty
intersection. We describe the structure of intersecting families of $r$-sets of
an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose
r-1}$ given by the Erd\H os-Ko-Rado Theorem.
  In particular, this extends the Hilton-Milner theorem on nontrivial
intersecting families and answers a recent question of Han and Kohayakawa for
large $n$. In the case $r=3$ we describe the structure of all intersecting
families with more than 10 edges. We also prove a stability result for the
Erdos matching problem. Our short proofs are simple applications of the
Delta-system method introduced and extensively used by Frankl since 1977.
