  A rack of order $n$ is a binary operation $\rack$ on a set $X$ of cardinality
$n$, such that right multiplication is an automorphism. More precisely,
$(X,\rack)$ is a rack provided that the map $x\mapsto x\rack y$ is a bijection
for all $y\in X$, and $(x\rack y)\rack z=(x\rack z)\rack (y\rack z)$ for all
$x,y,z\in X$. The paper provides upper and lower bounds of the form $2^{cn^2}$
on the number of isomorphism classes of racks of order $n$. Similar results on
the number of isomorphism classes of quandles and kei are obtained. The results
of the paper are established by first showing how an arbitrary rack is related
to its operator group (the permutation group on $X$ generated by the maps
$x\mapsto x\rack y$ for $y\in Y$), and then applying some of the theory of
permutation groups. The relationship between a rack and its operator group
extends results of Joyce and of Ryder; this relationship might be of
independent interest.
