Drillstring dynamics is the basis of safe and efficient drilling, and many researchers have made in-depth research on it with fruitful results. However, the safety of drill string is faced with severe challenges due to its extremely large slenderness ratio, harsh working environment and significant nonlinear characteristics, especially in extra-deep wells exceeding 10,000 meters. Therefore, it is essential to grasp the research process of drill string dynamics and extract the core key issues. Based on a brief review of the research progress of the three basic drill string vibration forms, this paper focused on the research progress of the coupled vibration of these three basic vibration forms, and discussed in detail the harm, formation mechanism, research methods, measurement techniques and control methods of two complex vibrations: stick-slip and whirl. In particular, the research of high frequency torsional oscillation (HFTO), which has attracted much attention in recent ten years, was summarized. Finally, preliminary conclusions and prospects were provided, aiming to offer useful references for the innovative development of drill string dynamics mechanisms and control methods. The existing research results showed that the whirl is the most complex vibration form of downhole drill string, including forward and backward regular whirl and irregular whirl motion, and most of them are irregular whirl motion. Among all the whirl modes, backward regular whirl is more harmful to the drill string, which will not only increase the fatigue damage of the drill tool, but also reduce the mechanical penetration rate of the bit. Stick-slip vibration is another highly harmful and complex form of vibration, which is induced by the interaction between the drill bit and the rock and can be well described by the velocity weakening effect. Stick-slip vibration is closely related to whirl motion. The maximum rotational speed in the slip stage of stick-slip vibration may reach 6-8 times of the ground rotational speed, accompanied with violent whirl motion. The high frequency torsional oscillation, whose vibration frequency is much higher than that of eddy and stick-slip vibration, is even more harmful, and the related mechanism is far from clear and is urgent to explore.