  We give estimates on the number $AL_H(x)$ of arithmetic lattices $\Gamma$ of
covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a
first concrete estimate on the number of arithmetic 3-manifolds of volume at
most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we
compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use
several different techniques: geometric (bounding the number of generators of
$\Gamma$ as a function of its covolume), number theoretic (bounding the number
of maximal such $\Gamma$) and sharp estimates on the character values of the
symmetric groups (to bound the subgroup growth of $\Gamma$).
