Nonlinear control methods have been efficient and promising techniques for the trajectory tracking of AUVs. Precise control for AUVs is the prerequisite to effectively execute underwater tasks and complete pre-planned missions without direct control from an operator. Various nonlinear control schemes have successfully addressed some of AUVs' complexities regarding uncertain model dynamics and unpredictable environmental disturbances. This has made AUVs efficient systems for ocean exploration for scientific, military, investigative, and industrial purposes, such as studying the ocean floor, payload delivery, time-critical strikes, and pipeline inspections. This paper presents a comprehensive review of past work and advances related to trajectory-tracking control schemes for AUVs, including fundamentals of nonlinear control and trajectory tracking in AUVs, nonlinear control methods for AUVs trajectory tracking, comparative analysis of nonlinear control methods, strengths, and weaknesses of different approaches in handling nonlinearities and uncertainties of AUV dynamics, and physical experiment studies. Future trends are also discussed to improve AUVs' stability, responsiveness, and accuracy in varying and uncertain conditions.