With the purpose of providing higher data rate and ultra-reliable and low-latency communications for the users, employing the small cells in the upcoming Fifth-Generation-New Radio (5G-NR) cellular networks and beyond is leading to the serious challenges in mobility management due to dynamicity of the user equipments (UEs). Among different issues related to the mobility of the UEs, the handover management is one of the key procedures to let the UEs experience a high quality of service (QoS)/quality of experience (QoE). So far, many protocols and algorithms have been proposed to enhance the mobility management in 5G-NR from various aspects, but still a thorough survey has not been incorporated to summarize the recent advances and future directions of the mentioned issue. Thus, the handover management and the corresponding challenges stand in the center of concentrations in this paper, with a glance on 4G to the advancements of 5G-NR. The challenges include QoS/QoE, throughput, delay, traffic load, the attacks during authentication process and resource allocation. To overcome the mentioned challenges, the handover procedure is evaluated by some key performance indicators such as handover ratio, handover failure, radio link failure and such like which depend on the received signal quality. The lack of new advancements and novel aspects of handover plus the separation of security and handover issues in previous works are perceived as research gaps and motivations for the current paper. In this regard, this paper aims to specify and analyze the technical issues, to provide an overview on the proposed methods and recent advances and to bring the future directions into the limelight. We categorize the concurrent standards and methods about the handover, and then, we survey the proposed algorithms including theoretical, algorithm-based and pattern-capturing approaches following the authentication process along with the vulnerabilities and the algorithms to counteract the attacks during handover. Also, we study various issues corresponding to network performance during the handover procedure. Finally, we discuss the open problems and future research directions.