In this paper, the flight control of the Unmanned aerial vehicle (UAV) is discussed with the proposed adaptive dynamic surface control method owing to its underactuated and non-linear characteristics. The proposed control algorithm is based on radial basis function (RBF) neural network and anti-saturation auxiliary system to realize high-precision trajectory tracking under time-varying disturbances and input saturation. First, the nonlinear dynamic model of the UAV with disturbances is established with the aid of rigid body motion theory. With the adoption of the dynamic surface control algorithm, the error surface and the Lyapunov function are defined to design the preliminary control law of the designed controller. Then the RBF neural network is introduced to estimate and compensate the disturbance. Further, an anti - saturation module is designed to tackle the problem of input saturation. By using the Lyapunov stability theory, it is proved that the stability and signal consistency of the closed-loop system are bounded, along with the constrained conditions of the control parameters. Simulation experiments have been performed and the results demonstrate that the proposed control algorithm has high-precision trajectory tracking ability and strong anti-disturbance capability under the input saturation constraint with high control performance.