In this article, we address the task of cooperative path following control of multiple autonomous quadcopters in the presence of unknown external disturbances. Under the assumption that the communications among the vehicles are bidirectional and continuous, a synchronized path following strategy is proposed that regulates the speed of each vehicle along its path to reach consensus in relative position. Moreover, collision-free transient paths from the vehicle initial positions to a group of suitable selected positions along the desired paths are designed. Building on the backstepping technique, the proposed path following controller for each individual vehicle drives the quadcopter toward, and to stay within an arbitrarily small neighborhood of its corresponding desired path, achieving global uniformly ultimately boundedness. In addition, the devised controller guarantees that the actuations are bounded with respect to the position error. The controller is also made robust to external constant or slowly time-varying disturbances by the introduction of dynamic estimators for the disturbances. A projection operator is used to ensure that the estimates remain within the prescribed bounds and are sufficiently smooth to be backstepped. In order to validate the effectiveness and performance of the proposed methodology, we present and analyze both simulation and experimental results.