The services based on Unmanned Aerial Vehicles (UAVs) have started to be used in many countries but not on a large scale yet. The paper describes the present status of UAV services and a concept of a solution for large-scale deployment of safe and reliable UAV services that use the 5G network for communication with UAVs. Based on the Standards Developing Organizations and related industry fora activities, unmanned traffic and airspace regulations, especially the commonly supported concept of Unmanned Aircraft Systems Traffic Management (UTM), UAV-related communication requirements and 5G System (5GS) features, we have developed an alternative approach to the integration of UAV and 5GS. The proposed concept includes a set of enablers that can be provided by Mobile Network Operators in order to not only support but also leverage UAV services. As work on many items is still in progress, we identify a list of open issues and challenges and present them at the end of the paper with the main focus on the MNO–UTM provider relationship and safe flights.