Abstract: A theoretical method for the analysis of the nonlinear elastic characteristics of curved blood vessels is presented. The curved blood vessels are modeled to be a circularly toroidal, incompressible, and elastic thick-walled tube with anisotropy and local triclinicity. On these assumptions, the deformations and strains of curved vessel walls are analyzed in detail, and a pseudo-strain energy function of 3-d exponential type is introduced to establish the constitutive relationships of nonlinear elasticity of curved vessel walls. The residual stresses and strains of curved vessel walls are studied. The way to generalize the constitutive equations obtained to involve residual stresses and strains is also discussed. These results can serve as the theoretical bases for the establishment of constitutive equations and the analysis of residual stresses and strains of curved vessel walls, as well as the numerical study of pulse wave propagation in curved blood vessels.