  This is an exposition of recent results -- obtained in joint work with
Andrzej Derdzinski -- on essentially conformally symmetric (ECS) manifolds,
that is, those pseudo-Riemannian manifolds with parallel Weyl curvature which
are not locally symmetric or conformally flat. In the 1970s, Roter proved that
while Riemannian ECS manifolds do not exist, pseudo-Riemannian ones do exist in
all dimensions $n\geq 4$, and realize all indefinite metric signatures. The
local structure of ECS manifolds is known, and every ECS manifold carries a
distinguished null parallel distribution $\mathcal{D}$, whose rank is always
equal to $1$ or $2$. We review basic facts about ECS manifolds, briefly discuss
the construction of compact examples, and outline the proof of a topological
structure result: outside of the locally homogeneous case and up to a double
covering, every compact rank-one ECS manifold is a bundle over $\mathbb{S}^1$
whose fibers are the leaves of $\mathcal{D}^\perp$. Finally, we mention some
classification results for compact rank-one ECS manifolds.
