This paper introduces a novel adaptive neural network-based sliding mode controller for trajectory tracking of a quadrotor UAV under unknown parameters such as inertia and mass. Unknown external disturbances and nonlinear aerodynamic forces are also considered. Due to its complex dynamics, the quadrotors need a dual controller for outer and inner loop control. For the commercial UAV, the inner loop controller, which ensures the attitude control, is usually assumed to be well developed and unmodifiable. Under this assumption, this paper focuses on the position control of a quadrotor UAV. A new position controller will be then designed and used to compute the appropriate input commands for the attitude controller to achieve the trajectory tracking task. The proposed method combines both Back-Propagation Neural Network (BPNN) scheme and sliding mode control to eliminate the effects of exogenous disturbances and model uncertainties. To illustrate the efficiency of the proposed controller, a comparative analysis is performed with different methods through experimental results.