With the widespread use of Permanent Magnet Synchronous Motor (PMSM) in industrial applications, sensorless PMSM has a major advantage over conventional PMSM. This work examines sensorless PMSM with adaptive observers, encompassing important methods such as the sliding mode observer (SMO), adaptive nonlinear extended state observer (ANLESO), Luenberger observer, and extended Kalman filter (EKF). These methods are investigated with the aim of achieving accurate observation and estimation of parameters such as rotor position in PMSM systems, and their effectiveness in solving the problems of insufficient torque and the high energy consumption is analyzed. The operational challenges of PMSM are briefly reviewed at the outset of this paper, and then the control strategies of the four previously stated methodologies are thoroughly described, with an emphasis on the benefits of these approaches in sensorless PMSM. The article concludes by summarising the research progress and future directions of these techniques, pointing out their potential to improve PMSM performance and stability.