Future integrated terrestrial, aerial, and space networks will involve thousands of low-Earth-orbit (LEO) satellites, which will form a network of mega-constellations. These megaconstellations will play a significant role in providing communication and Internet services anywhere, at any time, and for everything. Due to the large scale and highly dynamic nature of future LEO satellite networks (SatNets), their management will be a complicated process, especially the aspect of mobility management and its two components: location management and handover management. In this article, we present a comprehensive and critical review of the state-of-the-art research in location management for LEO SatNets. First, we give an overview of the Internet Engineering Task Force (IETF) mobility management standards (e.g., Mobile IPv6 and Proxy Mobile IPv6) and discuss the limitations of their location management techniques for future LEO SatNets. We highlight the mobility characteristics of future LEO SatNets and their challenging features, and we describe two unprecedented future location management scenarios. A taxonomy of existing location management solutions for LEO SatNets is also presented with solutions classified according to three approaches. The "Issues to consider" section draws attention to critical points related to each of the reviewed approaches that should be considered in future LEO SatNets location management. To identify the research gaps, the current state of LEO SatNets location management is summarized. Noteworthy future research directions are recommended. The article provides a road map for researchers and industry to shape the future of location management for LEO SatNets.