Enabling underwater wireless communications to conveniently interconnect various underwater deployed devices for data transmission, information sharing, and networking is among the most crucial issues in marine information networks. Apart from the conventional acoustic, optical, and electromagnetic techniques, the magnetic induction (MI) communication, as a promising alternative, has drawn ... [Show full abstract] significant attentions recently due to its inherent advantages in predictable channel responses, negligible propagation delay, and competitive energy consumption. In this article, we attempt to provide a comprehensive and in-depth survey of the existing research on underwater MI communications, classified as the four topics of channel modeling, reliability guarantee, range extension, and capacity enhancement, and present the state-of-the-art advances on each topic. Specifically, the approaches for channel modeling of underwater MI communications, including proposed channel models and involved basic theories, are first summarized. Then, the existing works on reliability guarantee are expounded following the evolution of antenna design from traditional single-directional to advanced multi-directional MI antennas. Furthermore, as to range extension, two typical categories of MI relaying techniques, i.e., the MI waveguide and the active relaying techniques, are reviewed in detail. In particular, the potential of a hybrid relay transmission scheme proposed by us through flexible combination of these two techniques in achieving both energy-efficient and long-range underwater transmission is also addressed in this part. Finally, the existing approaches on extending or reusing the available frequency bands to enhance the underwater MI channel capacity are elaborated. Challenges and open issues that need to be further investigated related to each of the four topics are also profoundly discussed and analyzed in each part.