With the growing popularity of Unmanned Aerial Vehicles (UAVs) for consumer applications, the number of accidents involving UAVs is also increasing rapidly. Therefore, motion safety of UAVs has become a prime concern for UAV operators. For a swarm of UAVs, a safe operation can not be guaranteed without preventing the UAVs from colliding with one another and with static and dynamically appearing, moving obstacles in the flying zone. In this paper, we present an online, collision-free path generation and navigation system for swarms of UAVs. The proposed system uses geographical locations of the UAVs and of the successfully detected, static and moving obstacles to predict and avoid: (1) UAV-to-UAV collisions, (2) UAV-to-static-obstacle collisions, and (3) UAV-to-moving-obstacle collisions. Our collision prediction approach leverages efficient runtime monitoring and Complex Event Processing (CEP) to make timely predictions. A distinctive feature of the proposed system is its ability to foresee potential collisions and proactively find best ways to avoid predicted collisions in order to ensure safety of the entire swarm. We also present a simulation-based implementation of the proposed system along with an experimental evaluation involving a series of experiments and compare our results with the results of four existing approaches. The results show that the proposed system successfully predicts and avoids all three kinds of collisions in an online manner. Moreover, it generates safe and efficient UAV routes, efficiently scales to large-sized problem instances, and is suitable for cluttered flying zones and for scenarios involving high risks of UAV collisions.