Gas foil bearing rotor (GFBR) systems have received significant interest in the field of rotordynamics and vibration analysis. GFBR systems have a wide range of high-speed turbomachinery applications. Due to the high speed, these machines are susceptible to rigorous vibration and instability. Gas foil bearings instigate large amplitude of vibrations at startup and shut down and severe subsynchronous motions during high-speed operations. Over the years, numerous work has been done in the field of high-speed rotors supported on gas foil bearings. Significant improvement has been observed in the stability and feasibility of the GFBR systems. However, accurate model predictions of gas foil bearing still remain a challenge for its widespread usage in high-speed turbomachinery. A comprehensive review needs to be done to study the previous work and pave the way for future research. The current review is majorly divided into three sections. Firstly, various models used for the performance prediction of gas foil bearings are compiled. After that, major causes of instability that manifest during the experiments and practice with gas foil bearing supported rotors is illustrated. Lastly, the developmental attempts made to inhibit the instability is summarised. This paper presents an overall picture of the current engineering scenario and future prospects of the GFBR systems.