Dynamic obstacle avoidance on quadrotors requires low latency. A class of sensors that are particularly suitable for such scenarios are event cameras. In this paper, we present a deep learning based solution for dodging multiple dynamic obstacles on a quadrotor with a single event camera and on-board computation. Our approach uses a series of shallow neural networks for estimating both the ego-motion and the motion of independently moving objects. The networks are trained in simulation and directly transfer to the real world without any fine-tuning or retraining. We successfully evaluate and demonstrate the proposed approach in many real-world experiments with obstacles of different shapes and sizes, achieving an overall success rate of 70% including objects of unknown shape and a low light testing scenario. To our knowledge, this is the first deep learning - based solution to the problem of dynamic obstacle avoidance using event cameras on a quadrotor. Finally, we also extend our work to the pursuit task by merely reversing the control policy, proving that our navigation stack can cater to different scenarios.