An adaptive control for a quadcopter aerial vehicle exposed to aggressive wind gusts is presented in this paper. The control scheme is composed by two parts; firstly, the attitude dynamics is robustly stabilized using a controller based on the quaternion formulation. Then, a translational flight control law with adaptive properties is designed using the sliding mode approach. The stability analysis of the whole system is proved using the Lyapunov theory. The performance of the closed-loop system is validated in real-time experiments and for validating the adaptive and robust properties of the controller, strong wind gusts are applied during flight tests. A video and some graphs, obtained from these experiments, illustrate the effectiveness and robustness of the proposed control strategy.