In this paper we study the response of bodies that are residually stressed within the context of a new class of constitutive relations, wherein the strains are assumed to be functions of the stresses. Such bodies are said to have residual stresses if there are stresses within the bodies even though the bodies are unstrained in the configuration of interest in the absence of external traction. Problems within the context of the norm of the gradient of the displacement field being small are considered, with regard to the determination of the residual stresses in an anisotropic cylindrical annulus with two preferred directions, and the nature of residual stresses within an anisotropic slab. The residual stresses in a body that is subject to incremental stresses are also studied.