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We give a new expression for the supercurrent and its conservation in curved${\cal N}=1$, $D=4$ superspace using the superconformal approach. The firstcomponent of the superfield, whose lowest component is the vector auxiliaryfield gives the (super)Einstein equations. Its trace and couplings to conformaland non-conformal matter is presented. In a suitable dilatational gauge, theconformal gauge, we obtain an update of the Callan-Coleman-Jackiw improvedcurrents for conformal matter, containing $R$-symmetry corrections for a newtraceless covariantly conserved energy--momentum tensor. We observe that in thePoincar\'e gauge, where standard Poincar\'e supergravity is usually formulated,the currents are not improved and then the higher conformal symmetry of thematter sector is obscured. The curvature multiplets are used to findsupersymmetric curved backgrounds and some examples are exhibited in agreementwith existing results.

The Supercurrent and Einstein equations in the Superconformal formulation