A conformally flat manifold is a manifold with a conformal class of Riemannian metrics containing, for each point
x
x
, a metric which is flat in a neighborhood of
x
x
. In this paper we classify closed conformally flat manifolds whose fundamental group (more generally, holonomy group) is nilpotent or polycyclic of rank
3
3
. Specifically, we show that such conformally flat manifolds are covered by either the sphere, a flat torus, or a Hopf manifold—in particular, their fundamental groups contain abelian subgroups of finite index. These results are applied to show that certain
T
2
{T^2}
-bundles over
S
1
{S^1}
(namely, those whose attaching map has infinite order) do not have conformally flat structures. Apparently these are the first examples of
3
3
-manifolds known not to admit conformally flat structures.