  Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic
hyperbolic manifolds of any given dimension. In dimension four and higher, we
show that there are about v^v such manifolds of volume at most v, considered up
to commensurability. Since the number of arithmetic ones tends to be
polynomial, almost all hyperbolic manifolds are non-arithmetic in an
appropriate sense. Our method involves a geometric graph-of-spaces construction
that relies on arithmetic properties of certain quadratic forms.
