Flexibility in power systems is ability to provide supply-demand balance, maintain continuity in unexpected situations, and cope with uncertainty on supply-demand sides. The new method and management requirements to provide flexibility have emerged from the trend towards power systems increasing renewable energy penetration with generation uncertainty and availability. In this study, the historical development of power system flexibility concept, the flexible power system characteristics, flexibility sources, and evaluation parameters are presented as part of international literature. The impact of variable renewable energy sources penetration on power system transient stability, small-signal stability, and frequency stability are discussed; the studies are presented to the researchers for further studies. Moreover, flexibility measurement studies are investigated, and methods of providing flexibility are evaluated. •What is flexibility in power systems?•Variable renewable energy penetration effects on power system.•Studies on power system flexibility measurement.•Solutions to provide power system flexibility.