Mobility is an essential factor in the 5G core network (CN). If the network control plane can predict mobility, the operation will be more efficient and agile with intelligent and proactive decisions. So far, mobility patterns and their prediction models have been extensively studied in the research community with cellular network datasets. Recently, 3GPP initiated the specification of a CN architecture for data analysis and machine learning. In this article, in accordance with this trend, we provide a taxonomy of mobility prediction frameworks in 5G CNs ranging from data collection to model serving, with consideration of the 3GPP architecture and interfaces; and we introduce two key use cases in 5G CNs, where the gains from mobility predictions are evaluated on datasets from live networks. In particular, one of the proposed methods, machine-learning-assisted adaptive paging, reduces signaling overhead by up to 75 percent.