  Let $G$ be a finite primitive permutation group and let $\kappa(G)$ be the
number of conjugacy classes of derangements in $G$. By a classical theorem of
Jordan, $\kappa(G) \geqslant 1$. In this paper we classify the groups $G$ with
$\kappa(G)=1$, and we use this to obtain new results on the structure of finite
groups with an irreducible complex character that vanishes on a unique
conjugacy class. We also obtain detailed structural information on the groups
with $\kappa(G)=2$, including a complete classification for almost simple
groups.
