In this chapter, using analytical approaches, we consider the problems of dynamics and stability of moving elastic rods and strings, axially traveling between two supports at a constant velocity. Transverse, longitudinal and torsional vibrations of the moving structure are reduced to the same mathematical form, a hyperbolic second-order partial differential equation. The analysis is then extended to the axially traveling string with damping. An analytical free-vibration solution is obtained. It is seen external friction leads to stabilization, whereas internal friction in the traveling material will destabilize the system in a dynamic mode at the static critical point. Finally, we consider the effects of bending rigidity, which in the case of paper materials introduces a singular perturbation to the governing equation. We consider an implicit exact eigensolution for beams, the effect of elastic supports at the boundaries to the vibration behavior of a long traveling beam, and the stability of a beam traveling in a homogeneous gravitational field.