  If two closed Jordan curves in the plane have precisely one point in common,
then it is called a {\em touching point}. All other intersection points are
called {\em crossing points}. The main result of this paper is a Crossing Lemma
for closed curves: In any family of $n$ pairwise intersecting simple closed
curves in the plane, no three of which pass through the same point, the number
of crossing points exceeds the number of touching points by a factor of at
least $\Omega((\log\log n)^{1/8})$.
  As a corollary, we prove the following long-standing conjecture of Richter
and Thomassen: The total number of intersection points between any $n$ pairwise
intersecting simple closed curves in the plane, no three of which pass through
the same point, is at least $(1-o(1))n^2$.
