The rapid growth of mobile data traffic will lead to the deployment of Ultra-Dense Networks (UDN) in the near future. Various networks must overlap to meet the massive demands of mobile data traffic, causing an increase in the number of handover scenarios. This will subsequently affect the connectivity, stability, and reliability of communication between mobile and serving networks. The inclusion of Unmanned Aerial Vehicles (UAVs)-based networks will create more complex challenges due to different mobility characterizations. For example, UAVs move in three-dimensions (3D), with dominant of line-of-sight communication links and faster mobility speed scenarios. Assuring steady, stable, and reliable communication during UAVs mobility will be a major problem in future mobile networks. Therefore, this study provides an overview on mobility (handover) management for connected UAVs in future mobile networks, including 5G, 6G, and satellite networks. It provides a brief overview on the most recent solutions that have focused on addressing mobility management problems for UAVs. At the same time, this paper extracts, highlights, and discusses the mobility management difficulties and future research directions for UAVs and UAV mobility. This study serves as a part of the foundation for upcoming research related to mobility management for UAVs since it reviews the relevant knowledge, defines existing problems, and presents the latest research outcomes. It further clarifies handover management of UAVs and highlights the concerns that must be solved in future networks.