The ability to converse with humans and follow commands in natural language is crucial for intelligent unmanned aerial vehicles (a.k.a. drones). It can relieve people's burden of holding a controller all the time, allow multitasking, and make drone control more accessible for people with disabilities or with their hands occupied. To this end, we introduce Aerial Vision-and-Dialog Navigation (AVDN), to navigate a drone via natural language conversation. We build a drone simulator with a continuous photorealistic environment and collect a new AVDN dataset of over 3k recorded navigation trajectories with asynchronous human-human dialogs between commanders and followers. The commander provides initial navigation instruction and further guidance by request, while the follower navigates the drone in the simulator and asks questions when needed. During data collection, followers' attention on the drone's visual observation is also recorded. Based on the AVDN dataset, we study the tasks of aerial navigation from (full) dialog history and propose an effective Human Attention Aided (HAA) baseline model, which learns to predict both navigation waypoints and human attention. Dataset and code will be released.