An elastically mounted circular cylinder, immersed in a cross-current and free to move along a rectilinear path, is subjected to vortex-induced vibrations (VIV). These vibrations develop through a mechanism referred to as lock-in, where body motion and vortex shedding synchronize at a frequency that may deviate both from the oscillator natural frequency and from the vortex shedding frequency past a fixed cylinder. The present numerical study aims at extending the analysis to curved trajectories, by considering that the cylinder is free to translate along a circular path. The Reynolds number based on the body diameter (D) and current velocity (U) is set to 100. A wide range of path radii, from 0.05D to 10D, and values of the reduced velocity (inverse of the oscillator natural frequency non-dimensionalized by D and U) up to 30 are examined, for the concave and convex configurations, i.e. the circular path centre located upstream or downstream of the cylinder. Path curvature results in a major alteration of the flow-body system behaviour compared with rectilinear VIV, with substantially different evolutions in the concave and convex configurations. In addition to the typical lock-in mechanism, two subharmonic forms of synchronization, at half and one third of vortex formation frequency, are uncovered in the convex configuration. They coexist with a desynchronized regime where the body and the flow oscillate at incommensurable frequencies. The four interaction regimes exhibit contrasted trends in terms of structural response, spatiotemporal organization of the wake and associated forces. They particularly differ by their symmetry properties, which are closely linked to the possible reconfiguration of the oscillator due to mean fluid forcing.