Autonomous underwater vehicles (AUVs) are one of the main tools to develop and utilize the ocean space for the mankind, and are playing an increasingly important role in the maritime domain. When the underwater vehicles are sailing, due to the complex and variable underwater environment, the systems generally exhibit poor control stability and weak anti-interference capabilities. To address this issue, this paper provides a systematic modeling and control of a representative AUV. Specifically, the models of the AUV kinetic and dynamics are first formulated, and the sophisticated three-dimensional motion is then decoupled into the corresponding horizontal and vertical planes, respectively. On this basis, we propose an Adaptive Sliding Mode Control (ASMC) strategy to realize stable control of the desired motion. Using Lyapunov theory, we demonstrate the stability of the closed-loop system. Subsequently, the corresponding numerical simulation experiments are conducted with different reference trajectories, and the results demonstrated that, compared to traditional proportional-integral-derivative (PID) and conventional sliding mode controllers, the proposed ASMC controller demonstrates superior performance with rapid convergence and outstanding anti-disturbance capability.