•Physical realizations and mechanical models of inerter are briefly presented.•Inerter-based vibration control systems and their applications are comprehensively reviewed.•The remaining challenges of inerter-based vibration control systems are discussed. Structural vibration control has received considerable research attentions in the past few decades, with special emphasis on developing effective, affordable and applicable control systems to protect the structures against natural or man-made hazards. In 2002, a two-terminal mechanical element, dubbed inerter, was proposed based on the force-current analogy, offering many potentials for upgrading conventional structural vibration control systems. Over the past two decades especially in the recent five years, extensive research efforts have been made for the development of inerter-based vibration control systems. This paper aims to provide a state-of-the-art review on the research and development of inerter-based passive vibration control systems and their applications. It begins with the concept and physical realizations of inerter. The mechanical models of inerter are then presented. Subsequently, this paper reviews the applications of inerter in civil engineering and discusses its benefits comparing to conventional vibration control systems. Finally, some discussions are made on the unresolved problems and the possible topics for future studies.