  In this paper we introduce the notion of Einstein-type structure on a
Riemannian manifold $\varrg$, unifying various particular cases recently
studied in the literature, such as gradient Ricci solitons, Yamabe solitons and
quasi-Einstein manifolds. We show that these general structures can be locally
classified when the Bach tensor is null. In particular, we extend a recent
result of Cao and Chen.
