Nowadays, there is a growing awareness on the social and economic importance of the ocean. In this context, being able to carry out a diverse range of operations underwater is of paramount importance for many industrial sectors as well as for marine science and to enforce restoration and mitigation actions. Underwater robots allowed us to venture deeper and for longer time into the remote and hostile marine environment. However, traditional design concepts such as propeller driven Remotely Operated Vehicles (ROVs), Autonomous Underwater Vehicles (AUVs), or tracked benthic crawlers, present intrinsic limitations, especially when a close interaction with the environment is required. An increasing number of researchers are proposing legged robots as a bioinspired alternative to traditional designs, capable of yielding versatile multi-terrain locomotion, high stability, and low environmental disturbance. In this work, we aim at presenting the new field of underwater legged robotics in an organic way, discussing the prototypes in the state-of-the-art and highlighting technological and scientific challenges for the future. First, we will briefly recap the latest developments in traditional underwater robotics from which several technological solutions can be adapted, and on which the benchmarking of this new field should be set. Second, we will the retrace the evolution of terrestrial legged robotics, pinpointing the main achievements of the field. Third, we will report a complete state of the art on Underwater Legged Robots (ULRs) focusing on the innovations with respect to the interaction with the environment, sensing and actuation, modelling and control, and autonomy and navigation. Finally, we will thoroughly discuss the reviewed literature by comparing traditional and legged underwater robots, highlighting interesting research opportunities, and presenting use case scenarios derived from marine science applications.