Drones have attracted extensive attention for their environmental, civil, and military applications. Because of their low cost and flexibility in deployment, drones with communication capabilities are expected to play key important roles in Fifth Generation (5G), Sixth Generation (6G) mobile networks, and beyond. 6G and 5G are intended to be a full-coverage network capable of providing ubiquitous connections for space, air, ground, and underwater applications. Drones can provide airborne communication in a variety of cases, including as Aerial Base Stations (ABSs) for ground users, relays to link isolated nodes, and mobile users in wireless networks. However, variables such as the drone’s free-space propagation behavior at high altitudes and its exposure to antenna sidelobes can contribute to radio environment alterations. These differences may render existing mobility models and techniques as inefficient for connected drone applications. Therefore, drone connections may experience significant issues due to limited power, packet loss, high network congestion, and/or high movement speeds. More issues, such as frequent handovers, may emerge due to erroneous transmissions from limited coverage areas in drone networks. Therefore, the deployments of drones in future mobile networks, including 5G and 6G networks, will face a critical technical issue related to mobility and handover processes due to the main differences in drones’ characterizations. Therefore, drone networks require more efficient mobility and handover techniques to continuously maintain stable and reliable connection. More advanced mobility techniques and system reconfiguration are essential, in addition to an alternative framework to handle data transmission. This paper reviews numerous studies on handover management for connected drones in mobile communication networks. The work contributes to providing a more focused review of drone networks, mobility management for drones, and related works in the literature. The main challenges facing the implementation of connected drones are highlighted, especially those related to mobility management, in more detail. The analysis and discussion of this study indicates that, by adopting intelligent handover schemes that utilizing machine learning, deep learning, and automatic robust processes, the handover problems and related issues can be reduced significantly as compared to traditional techniques.