  This is an exposition of some recent results on ECS manifolds, by which we
mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither
conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS
metrics are indefinite. We state two classification theorems, describing the
local structure of ECS manifolds, and outline an argument showing that compact
ECS manifolds exist in infinitely many dimensions greater than 4. We also
discuss some properties of compact manifolds that admit ECS metrics, and
provide a list of open questions about compact ECS manifolds.
