The Internet of Underwater Things (IoUT) is an emerging communication ecosystem developed for connecting underwater objects in maritime and underwater environments. The IoUT technology is intricately linked with intelligent boats and ships, smart shores and oceans, automatic marine transportations, positioning and navigation, underwater exploration, disaster prediction and prevention, as well as with intelligent monitoring and security. The IoUT has an influence at various scales ranging from a small scientific observatory, to a mid-sized harbor, and to covering global oceanic trade. The network architecture of IoUT is intrinsically heterogeneous and should be sufficiently resilient to operate in harsh environments. This creates major challenges in terms of underwater communications, whilst relying on limited energy resources. Additionally, the volume, velocity, and variety of data produced by sensors, hydrophones, and cameras in IoUT is enormous, giving rise to the concept of Big Marine Data (BMD), which has its own processing challenges. Hence, conventional data processing techniques will falter, and bespoke Machine Learning (ML) solutions have to be employed for automatically learning the specific BMD behavior and features facilitating knowledge extraction and decision support. The motivation of this article is to comprehensively survey the IoUT, BMD, and their synthesis. It also aims for exploring the nexus of BMD with ML. We set out from underwater data collection and then discuss the family of IoUT data communication techniques with an emphasis on the state-of-the-art research challenges. We then review the suite of ML solutions suitable for BMD handling and analytics. We treat the subject deductively from an educational perspective, critically appraising the material surveyed. Accordingly, the reader will become familiar with the pivotal issues of IoUT and BMD processing, whilst gaining an insight into the state-of-the-art applications, tools, and techniques. Finally, we analyze the architectural challenges of the IoUT, followed by proposing a range of promising direction for research and innovation in the broad areas of IoUT and BMD. Our hope is to inspire researchers, engineers, data scientists, and governmental bodies to further progress the field, to develop new tools and techniques, as well as to make informed decisions and set regulations related to the maritime and underwater environments around the world.