In practical applications for aerial transportation tasks, owing to the complexity of the environment, the external disturbances are uncertain, which will affect the performance of the control system. This paper studies the control problem of the quadrotor unmanned aerial vehicle (UAV) transportation system with unknown external disturbances. A novel adaptive neural network (NN) control law, based on an artificially constructed sliding manifold and radial basis functions (RBF) NNs, is proposed to achieve stable positioning of the quadrotor without any linearizing operations. Specifically, RBFNN approximators are utilized to compensate for uncertainties/disturbances, and the update law for weights of neural network is designed. By Lyapunov techniques, the stability of system is guaranteed and all the signals in the closed-loop system are proved to be bounded. Simulation results validate the superior performance and robustness of the proposed controller by comparing it with the classical control scheme.