Stabilized space}time "nite-element methods are employed to investigate vortex-induced vibrations of a light circular cylinder placed in a uniform #ow at Reynolds number in the range of 10}10. The governing equations for the #uid #ow are the Navier}Stokes equations for incompressible #ows. The cylinder is mounted on lightly damped, #exible supports and allowed to vibrate, both in the in-line and cross-#ow directions under the action of aerodynamic forces. Results are presented for various values of the structural frequency of the oscillator including those that are super-harmonics of the vortex-shedding frequency for a stationary cylinder. In certain cases the e!ect of the mass of the oscillator is also examined. The motion of the cylinder alters the #uid #ow signi"cantly. To investigate the long-term dynamics of the non-linear oscillator, beyond the initial transient solution, long-time integration of the governing equations is carried out. For e$cient utilization of the available computational resources the non-linear equation systems, resulting from the "nite-element discretization of the #ow equations, are solved using the preconditioned generalized minimal residual (GMRES) technique. Flows at lower Reynolds numbers are associated with organized wakes while disorganized wakes are observed at higher Reynolds numbers. In certain cases, competition is observed between various modes of vortex shedding. The #uid}structure interaction shows a signi"cant dependence on the Reynolds number in the range that has been investigated in this article. In certain cases lock-in while in some other cases soft-lock-in is observed. The trajectory of the cylinder shows very interesting patterns including the well-known ΒΈissajou "gure of 8. Several mechanisms of the non-linear oscillator for self-limiting its vibration amplitude are observed.