The geometrically nonlinear problem for a system of rigid bodies, equipped with elastic devices, is formulated. A general, nonlinear, strain-displacement function is introduced, and equilibrium enforced in the current configuration, taken at a finite distance from the reference one. Elasticity is assumed linear. By following the direct approach, the balance equations are derived. Successively, a virtual displacement field is assigned starting from the current configuration and duality properties among the displacement-dependent congruence and equilibrium operators are shown. These latter lead to assess the validity of the Virtual Work Principle for (true) finite displacements. Finally, the Total Potential Energy Theorem is proved to hold also in the nonlinear field; it represents an useful tool to derive nonlinear balance equations, even in case of constrained variational problems. The second part of the chapter is dedicated to applications. To limit the algebra, only single-DOF systems are considered. First, the post-buckling behavior is studied, and the stability of the bifurcated path is investigated. Bifurcation of pitchfork and transcritical types are detected. Then, the free and forced motions of an undamped oscillator are analyzed via the Harmonic Balance Method, and the frequency-amplitude response is built-up, close to the resonance. Finally, damped systems loaded by velocity-dependent non-conservative forces are addressed, and the limit-cycle is described as a function of the load.