The linear and nonlinear theories of elastic slender curved rods are formulated in a systematic manner. Equations of equilibrium for stress resultants and moments are derived. The generalized strains are defined based on the principle of virtual work. Constitutive equations corresponding to small strains are obtained. The field equations for finite deformations of curved rods can be simplified in the case of small axial strain and moderately small rotations. Further simplifications can be made for the case of slightly curved rods. Two examples are presented to illustrate applications of the developed theories.