Power systems are evolving to the networks with proliferated penetration of renewable energy resources to leverage their environmental and economic advantages. However, due to the stochastic nature of renewables, the management of the rapidly increasing uncertainty and variability in power system planning and operation is of crucial significance. This paper represents a comprehensive overview of power system flexibility as an effective way to maintain the power balance at every moment. Definitions of power system flexibility from various aspects are explained to reach the reliable and economic planning and operation of the power system. The effects of the high penetration of variable energy resources on power systems and the evolution of flexibility in response to renewables are studied. A variety of resources during the flexibility evolutionary transition are introduced and discussed. As an influential flexibility solution in current power systems integrated with renewable resources, market design improvement is widely reviewed in this paper, and required modifications in market design mechanisms are investigated pertaining to various time horizons.