This is a comprehensive review about fluid-structure interaction of an actively or passively rotating cylinder taking into account flow-induced vibration (FIV). Four response modes are summarized: active rotation, passive rotation, and co-existence of one kind of rotation and FIV. For active rotation response, the impact factors including Reynolds number, rotating velocity and frequency, and occurrence of FIV are examined. Generally, the active rotation can boost drag-reduction but has a negative effect on lift force, as a cylinder rotating faster is more likely to form a single-sided wake structure rather than classical Kármán vortex street. More unique wake modes such as T+S (a triplet of vortices and a single vortex alternately shed in one cycle), D+ and D− (two layers of vorticity in opposite signs corresponding to the positive or negative drag force), and C(AS) (the coalescence of small asymmetric vortices, together with its wake switching) appear when active rotation and FIV are considered simultaneously. In contrast, the vibration response of a cylinder with a time-varying rotation velocity would be suppressed at specific rotation amplitudes and frequencies. For flow-induced rotation (FIR) without vibration, the rotation behavior mainly depends on the cylinder shape and Reynolds number. Although the examples of coexisting FIR and FIV such as offshore wind turbine are extremely common in engineering, relevant systematic investigations are scanty, even for a simple structure like square and triangle. Moreover, previous literature is mainly concerned about the effect of FIR to FIV, the opposite studies have not been reported, yet.