This paper studies path-following control of underactuated autonomous underwater vehicle (AUV) with parameter perturbation and time-varying disturbances. First, the three-dimensional path following error equation is established through the Serret-Frenet frame. Then, the kinematics guidance laws of underactuated AUV are established by backstepping method. And the dynamic controller is designed through nonsingular terminal sliding mode control (NTSMC) method and extended disturbance observer (EDO), so that AUV can keep up with the desired velocity on external disturbances. Finally, it is proved that the designed path-following controller can make the error converge globally asymptotically. Simulation shows that this method can accomplish the task of path tracking control well.