In this paper, research on transverse vibrations of axially moving strings and their control is thoroughly reviewed. In the last few decades, there have been extensive studies on analysis and control of transverse vibrations of axially moving strings because of the wide applications of many engineering devices that axially moving strings represent. In the investigations adopting linear models of moving strings, the paper summarizes recent studies on modal analysis, complicatedly constrained strings, coupled vibrations, and parametric vibration, as well as some early results. In the investigations adopting non-linear models of moving strings, the paper presents the governing equations with large amplitude, and reviews progress on discretized or direct approximate analytical analyses and numerical approaches based on the Galerkin method or the finite difference method. Furthermore, investigations are reported on modeling of damping mechanisms as vis-coelastic materials, coupled vibration of power transmission systems, and bifurcation and chaos. The state of the art of active control of moving strings is surveyed on controlla-bility and observability, the Laplace transform domain analysis and the energy analysis,nonlinear vibration control and adaptive vibration control. Finally, future research direc-tions are suggested such as nonlinear vibration of moving strings under complex con-straints and couplings, energetics of nonlinear and time-varying strings, bifurcation and chaos in transverse motion of moving strings, control of hybrid systems containing moving strings, robust and adaptive controls of nonlinear mov-ing strings, and experimental investigations. In this review article there are 242 references cited.