  We investigate the structure of conformal $C$-spaces,a class of Riemmanian
manifolds which naturally arises as aconformal generalisation of the Einstein
condition. A basic question is when such a structure is closed, or equivalently
locally conformally Cotton. In dimension 4 we obtain a full answer to this
question and also investigate the incidence of the Bach condition on this class
of metrics. This is related to earlier results obtained in the Einstein-Weyl
context.
