A wave-induced motion compensation crane and corresponding vessel and method are disclosed. The crane includes a motion compensation device at a tip end portion of the boom structure to compensate for X-Y wave-induced motion and a heave compensation device for Z-motion. The motion compensation device includes a moveable jib beam that extends in a substantially horizontal direction. The jib beam is slewable about a substantially vertical slew axis and translatable in a longitudinal direction of the jib beam. Preferably, the jib beam can be levelled based on the angular orientation of the boom structure.