Stay　cables　are　one　of　the　key　elements　of　cable-stayed　bridges　and　are　characterized　by　lightweight,　low　inherent　damping,　and　high　flexibility.　They　are　continuously　subjected　to　small-to　large-amplitude　vibrations　due　to　various　types　of　dynamic　loads　that　may,　in　the　long　term,　cause　fatigue　and　fracture　problems　for　the　cable　system,　and　may　eventually　compromise　the　safety　of　cable-stayed　bridges.　Thus,　several　countermeasures　including　surface　profiling,　cross-ties,　and　structural　vibrational　control　systems　have　been　used　to　improve　the　dynamic　performance　of　stay　cables.　This　article　presents　a　comprehensive　state-of-the-art　and　state-of-the-practice　review　of　structural　vibration　control　systems　specifically　designed　and　used　for　the　cables　in　cable-stayed　bridges.　Generally,　the　stay　cable　dampers　are　classified　as　internal　and　external　dampers.　Consequently,　important　aspects　of　each　control　strategy　are　highlighted　and　various　types　of　devices　and　their　designs　are　discussed　to　find　the　best　control　solution　for　suppressing　the　cable　vibrations.　©　2021,　CIMNE,　Barcelona,　Spain.