Autonomous Underwater Vehicles (AUVs) are playing an increasingly prominent role in the field of marine scientific research. Control methods and path planning algorithms, as the core technologies of AUVs, have a direct impact on their application prospects and are of great significance and worth studying. The purpose of this paper is to discuss and outline the development of control methods and path planning algorithms for AUVs in the last five years. Firstly, this paper describes the fundamentals, characteristics, as well as advantages and disadvantages of several typical control methods and several types of representative path-planning algorithms, surveys related work, and presents the latest research progress in this field. Secondly, this paper proposes some promising future research directions that are worth studying in the field of AUVs and provides guidance for its application and development.