  By ECS manifolds one means pseudo-Riemannian manifolds of dimensions
$\,n\ge4\,$ which have parallel Weyl tensor, but not for one of the two obvious
reasons: conformal flatness or local symmetry.
  As shown by Roter [10, 2], they exist for every $\,n\ge4$, and their metrics
are always indefinite. The local structure of ECS manifolds has been completely
described [3].
  Every ECS manifold has an invariant called rank, equal to 1 or 2. Known
examples of compact ECS manifolds [4, 6], representing every dimension
$\,n\ge5$, are of rank 1. When $\,n\,$ is odd, some further, recently found
examples are locally homogeneous [7].
  We outline the proof of the author's result, joint with Ivo Terek [5], which
states that a compact rank-one ECS manifold, if not locally homogeneous,
replaced if necessary by a two-fold isometric covering, must be the total space
of a bundle over the circle.
