In this article, we propose a solution to the problem of path following for a quadcopter aircraft with unknown vehicle parameters (mass and moment of inertia) and external disturbances. By employing the backstepping technique, the proposed adaptive control strategy guarantees the following: the quadcopter is globally steered toward, and kept within, an arbitrarily small neighborhood of a desired smooth path, achieving global uniformly ultimately boundedness; compared to trajectory tracking, a smoother convergence is obtained as the control actuation signals (thrust force and torque) are bounded with respect to the position error, and the designed timing law ensures that the desired path starts to move only when the vehicle gets close to the desired path; and a single adaptive control law can be used for accurate motion control of aerial vehicles with a wide range of inertial properties, without the need for retuning control gains or other parameters. Moreover, the controller is also made robust to external constant and slowly time-varying disturbances through the design of disturbance estimators. To demonstrate the effectiveness and performance of the proposed control strategies, simulation and experimental results are presented and analyzed.