  A nonlocal version of Einstein-Gauss Bonnet (EGB) gravity can be generated by
a procedure that follows closely the steps of the derivation of the conformal
anomaly effective action. The action is obtained using a Weyl decomposition of
the metric, with the conformal factor removed by a finite renormalization of
the topological term. We outline the mains steps in the derivation of this
action, stressing on the analogies and differences respect to the anomaly
action and to the ordinary $4d$ EGB theory formulated as a special version of
dilaton gravity. These formulations allow a systematic investigation of the
nonlocal $R\Box^{-1}$ corrections to General Relativity. In the case of
conformal anomaly action they are motivated by the generation of a conformal
backreaction, due to the breaking of conformal symmetry in the early universe.
