The key enablers for emerging cellular networks such as densification, concurrent operation at multiple bands and harnessing mmWave spectrum give birth to a peculiar set of new network management challenges. One such key challenge is the user mobility management. In this article, we identify the key issues that render current mobility management paradigm inadequate for delivering the expected Quality of Experience (QoE) and resource efficiency in emerging and future cellular networks. Together, these challenges call for paradigm shift in the way mobility is managed in cellular networks. We present an Advanced Mobility Management and Utilization Framework (A-MMUF) that can enable this paradigm shift by transforming mobility management from being a reactive to a proactive process. The core idea of A-MMUF is to build upon Mobility Prediction Models (MPMs) to predict various attributes of user mobility and traffic patterns such as next candidate cell for handover (HO), time of HO, future cell loads. These predictions are then leveraged to not only improve HO process for better QoE and less signaling overhead, but also to enable proactive automation to further maximize network performance in terms of energy and spectrum efficiency. However, understandably, the gains offered by the A-MMUF hinge on the accuracy of the MPMs, which may vary, not only with the choice of underlying machine learning technique, but also with the volume, variety, and fidelity of the training data. We analyze the potential gains of A-MMUF through three case studies: namely proactive HO, proactive Mobility Load Balancing (P-MLB), and proactive Energy Savings (P-ES). In addition to demonstrating significant gains in all three case studies, the results provide useful insights into the agility vs accuracy tradeoff that can be leveraged for choosing optimal machine learning models for practical deployment of A-MMUF.