Enabling users to move to different geographical locations within a network and still be able to maintain their connectivity and most essentially, continuity of service, is what makes any wireless network ubiquitous. Whilst challenging, modern day wireless networks, such as 3GPP-LTE, provision satisfactory mobility management (MM) performance. However, it is estimated that the number of mobile subscriptions will approximately touch 9 billion and the amount of data traffic will expand by 5 times in 2024 as compared to 2018. Further, it is expected that this trend of exponential growth will be maintained well into the future. To cope with such an exponential increase in cellular traffic and users alongside a burgeoning demand for higher Quality of Service (QoS), the future networks are expected to be highly dense and heterogeneous. This will severely challenge the existing MM solutions and ultimately render them ineffective as they will not be able to provide the required reliability, flexibility, and scalability. Consequently, to serve the 5G and beyond 5G networks, a new perspective to MM is required. Hence, in this article we present a novel discussion of the functional requirements from MM strategies for these networks. We then provide a detailed discussion on whether the existing mechanisms conceived by standardization bodies such as IEEE, IETF, 3GPP (including the newly defined 5G standards) and ITU, and other academic and industrial research efforts meet these requirements. We accomplish this via a novel qualitative assessment, wherein we evaluate each of the discussed mechanisms on their ability to satisfy the reliability, flexibility and scalability criteria for future MM strategies. We then present a study detailing the research challenges that exist in the design and implementation of MM strategies for 5G and beyond networks. Further, we chart out the potential MM solutions and the associated capabilities they offer to tackle the persistent challenges. We conclude this paper with a vision for the 5G and beyond MM mechanisms.