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The growing use of Unmanned Aerial Vehicles (UAVs) for various applications requires ubiquitous and reliable connectivity for safe control and data exchange between these devices and ground terminals. Depending on the application, UAV-mounted wireless equipment can either be an aerial user equipment (AUE) that co-exists with the terrestrial users, or it can be a part of wireless infrastructure providing a range of services to the ground users. For instance, AUE can be used for real-time search and rescue and Aerial Base Station (ABS) can enhance coverage, capacity and energy efficiency of wireless networks. In both cases, UAV-based solutions are scalable, mobile, fast to deploy. However, several technical challenges have to be addressed. In this work, we present a tutorial on wireless communication with UAVs, taking into account a wide range of potential applications. The main goal of this work is to provide a complete overview of the main scenarios (AUE and ABS), channel and performance models, compare them, and discuss open research points. This work gives a comprehensive overview of the research done until now and depicts a comprehensive picture to foster new ideas and solutions while avoiding duplication of past work. We start by discussing the open challenges of wireless communication with UAVs. To give answers to the posed questions, we focus on the UAV communication basics, mainly providing the necessary channel modeling background and giving guidelines on how various channel models should be used. Next, theoretical, simulation- and measurement-based approaches, to address the key challenges for AUE usage, are presented. Moreover, in this work, we aim to provide a comprehensive overview on how UAV-mounted equipment can be used as a part of a communication network. Based on the theoretical analysis, we show how various network parameters can be optimized., Comment: 42 pages, 32 Figures