This paper summarizes current understanding of the processes that determine the dynamics of the subsea permafrost–hydrate system existing in the largest, shallowest shelf in the Arctic Ocean; the East Siberian Arctic Shelf (ESAS). We review key environmental factors and mechanisms that determine formation, current dynamics, and thermal state of subsea permafrost, mechanisms of its destabilization, and rates of its thawing; a full section of this paper is devoted to this topic. Another important question regards the possible existence of permafrost-related hydrates at shallow ground depth and in the shallow shelf environment. We review the history of and earlier insights about the topic followed by an extensive review of experimental work to establish the physics of shallow Arctic hydrates. We also provide a principal (simplified) scheme explaining the normal and altered dynamics of the permafrost–hydrate system as glacial–interglacial climate epochs alternate. We also review specific features of methane releases determined by the current state of the subsea-permafrost system and possible future dynamics. This review presents methane results obtained in the ESAS during two periods: 1994–2000 and 2003–2017. A final section is devoted to discussing future work that is required to achieve an improved understanding of the subject.