The object of this paper is to develop a finite displacement theory of naturally curved and twisted rods undergoing finite rotations. Particular attention is paid to investigate the coupling of finite rotations in space under the Bernoulli-Euler hypothesis. A finite rotation vector is employed to derive the displacement field available for finite rotations. A new variable is introduced as a fourth parameter associated with rotations of cross sections. Then the twist and curvatures after the deformation are expressed in terms of four parameters without using small-strain assumptions. The equilibrium equations and the associated boundary conditions, in which second order terms with respect to displacement components are fully taken into account, are derived from the principle of virtual work. The accuracy of the present equilibrium equations are confirmed through comparisons with those obtained by the equilibrium method.