Autonomous micro aerial vehicles (MAVs) have cost and mobility benefits, making them ideal robotic platforms for applications including aerial photography, surveillance, and search and rescue. As the platform scales down, MAVs become more capable of operating in confined environments, but it also introduces significant size and payload constraints. A monocular visual-inertial navigation system (VINS), consisting only of an inertial measurement unit (IMU) and a camera, becomes the most suitable sensor suite in this case, thanks to its light weight and small footprint.In fact, it is the minimum sensor suite allowing autonomous flight with sufficient environmental awareness. In this paper, we show that it is possible to achieve reliable online autonomous navigation using monocular VINS. Our system is built on a customized quadrotor testbed equipped with a fisheye camera, a low-cost IMU, and heterogeneous onboard computing resources. The backbone of our system is a highly accurate optimization-based monocular visual-inertial state estimator with online initialization and self-extrinsic calibration. An onboard GPU-based monocular dense mapping module that conditions on the estimated pose provides wide-angle situational awareness. Finally, an online trajectory planner that operates directly on the incrementally built threedimensional map guarantees safe navigation through cluttered environments. Extensive experimental results are provided to validate individual system modules as well as the overall performance in both indoor and outdoor environments.