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The two-loop anomalous dimension of the chiral matter superfields iscalculated for a general ${\cal N}=1$ supersymmetric gauge theory regularizedby higher covariant derivatives. We obtain both the anomalous dimension definedin terms of the bare couplings, and the one defined in terms of therenormalized couplings for an arbitrary renormalization prescription. For theone-loop finite theories we find a simple relation between the higherderivative regulators under which the anomalous dimension defined in terms ofthe bare couplings vanishes in the considered approximation. In this case theone-loop finite theory is also two-loop finite in the HD+MSL scheme. Using theassumption that with the higher covariant derivative regularization the NSVZequation is satisfied for RGFs defined in terms of the bare couplings, weconstruct the expression for the three-loop $\beta$-function. Again, the resultis written both for the $\beta$-function defined in terms of the bare couplingsand for the one defined in terms of the renormalized couplings for an arbitraryrenormalization prescription.

Two-loop renormalization of the matter superfields and finiteness of ${\cal N}=1$ supersymmetric gauge theories regularized by higher derivatives