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The large-scale mobile underwater wireless sensor network (UWSN) is a novel networking paradigm to explore aqueous environments. However, the characteristics of mobile UWSNs, such as low communication bandwidth, large propagation delay, floating node mobility, and high error probability, are significantly different from ground-based wireless sensor networks. The novel networking paradigm poses interdisciplinary challenges that will require new technological solutions. In particular, in this article we adopt a top-down approach to explore the research challenges in mobile UWSN design. Along the layered protocol stack, we proceed roughly from the top application layer to the bottom physical layer. At each layer, a set of new design intricacies is studied. The conclusion is that building scalable mobile UWSNs is a challenge that must be answered by interdisciplinary efforts of acoustic communications, signal processing, and mobile acoustic network protocol design.