Autonomous underwater vehicles (AUVs) are playing an ever-growing role in modern subocean operations, generating a demand for faster, more manoeuvrable designs capable of deployments of increasingly longer durations. In order to meet these demands, vehicle developers have been looking to biological aquatic animals for inspiration. After evolving for millions of years, fish and cetaceans have developed fast efficient locomotion techniques, with levels of manoeuvrability that far outperform conventional engineered marine locomotion systems. This paper aims to give a brief introduction into some of the biologically inspired propulsion mechanisms that have been developed, to explain their strengths, their weaknesses, and the motivation behind them, and then finally to predict future trends in biomimetic AUV propulsion design.