The past years have shown a remarkable growth in use-cases for micro aerial vehicles (MAVs). Conceivable indoor applications require highly robust environment perception, fast reaction to changing situations, and stable navigation, but reliable sources of absolute positioning like GNSS or compass measurements are unavailable during indoor flights. We present a high-performance autonomous inventory MAV for operation inside warehouses. The MAV navigates along warehouse aisles and detects the placed stock in the shelves alongside its path with a multimodal sensor setup containing an RFID reader and two high-resolution cameras. We describe in detail the SLAM pipeline based on a 3D lidar, the setup for stock recognition, the mission planning and trajectory generation, as well as a low-level routine for avoidance of dynamical or previously unobserved obstacles. Experiments were performed in an operative warehouse of a logistics provider, in which an external warehouse management system provided the MAV with high-level inspection missions that are executed fully autonomously.