The Internet of Drones (IoD) recently gained momentum due to its high adaptability to a wide variety of complex scenarios. Indeed, Unmanned Aerial Vehicles (UAVs) can successfully be employed in different applications, thanks to some technological and practical advantages: high mobility, capability to extend wireless coverage areas, or ability to reach places inaccessible to humans. Moreover, the employment of drones promisingly improves the performance parameters of different network architectures. Nevertheless, the adoption of networks of drones gives rise to several issues related to the unreliability of the wireless medium, the duration of batteries, and the high mobility degree, which may cause frequent topology changes. Also security and privacy issues need to be properly investigated. With respect to other surveys on IoD-related topics, the goal of the present work is to categorize the multifaceted aspects of IoD, proposing a classification approach of the IoD environment that develops along two main directions. At a macroscopic level, it follows the structure of the Internet protocol stack, starting from the physical layer and extending to the upper layers, without neglecting cross-layer and optimization approaches. At a finer level, all the most relevant works belonging to each layer of the stack are further classified, according to the different issues peculiar of the layer, and highlighting the most relevant differences with the other surveys present in literature. To provide a deeper insight in the theme, the present work embraces many facets of the IoD, including privacy and security considerations as well as the potential economic impact of the IoD. Finally, a discussion on the main research challenges and possible future directions is carried out, focusing on the open issues and the most promising technologies that deserve to be further developed in the IoD field.