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In this paper, we describe our autonomous vision-based quadrotor MAV system which maps and explores unknown environments. All algorithms necessary for autonomous mapping and exploration run on-board the MAV. Using a front-looking stereo camera as the main exteroceptive sensor, our quadrotor achieves these capabilities with both the Vector Field Histogram+ (VFH+) algorithm for local navigation, and the frontier-based exploration algorithm. In addition, we implement the Bug algorithm for autonomous wall-following which could optionally be selected as the substitute exploration algorithm in sparse environments where the frontier-based exploration under-performs. We incrementally build a 3D global occupancy map on-board the MAV. The map is used by the VFH+ and frontier-based exploration in dense environments, and the Bug algorithm for wall-following in sparse environments. During the exploration phase, images from the front-looking camera are transmitted over Wi-Fi to the ground station. These images are input to a large-scale visual SLAM process running off-board on the ground station. SLAM is carried out with pose-graph optimization and loop closure detection using a vocabulary tree. We improve the robustness of the pose estimation by fusing optical flow and visual odometry. Optical flow data is provided by a customized downward-looking camera integrated with a microcontroller while visual odometry measurements are derived from the front-looking stereo camera. We verify our approaches with experimental results.