The existing literature explores the numerous benefits of UAV usage in modern cellular networks, including optimal resource allocation, maximizing vendor profit margins, and improving throughput. Every day a modern person uses various mobile devices to transmit and perceive digital information. In this regard, the demands for the quality of communication provided to subscribers, its availability and speed are growing. This issue is a key issue for the fifth generation networks (5G, Fifth-generation) and subsequent generations (B5G, Beyond 5G). UAVs (Unmanned Aerial Vehicles) are expected to become an important component of new generations of wireless networks that could potentially facilitate wireless deployment and support high-speed data transmission everywhere. Compared to fixed networks of fixed infrastructure, UAVs have such important attributes as: deployment flexibility, line-of-sight reliability, additional degrees of freedom in deployment, and controlled mobility. In this paper we present a systematic review of the UAV usage in modern cellular networks, as well as an overview of an integrated space-air-ground network, together with a description of the problems of deploying UAV networks. The above-mentioned reasons grant the timeliness and motivation of the review under consideration.