Power may be extracted from a flowing fluid in a variety of ways. Turbines using one or more oscillating foils are under increasingly active investigation, as an alternative to rotary wind turbines and river, oceanic and tidal current water turbines, although industrial development is at a very nascent stage. Such flapping foil turbines promise some key potential advantages, including lower foil velocities (and hence lower noise and wildlife impact), and more effective small-scale and shallow water operation. The role of a number of parameters is investigated, including foil kinematics (modes, frequencies, amplitudes and time histories of motion), foil and system geometry (shape, configuration and structural flexibility), and flow physics effects (Reynolds number and turbulence, shear flows and ground effect). Details of the kinematics are shown to have the single largest influence on power output and efficiency (measured as the ratio of power output to that available and accessible in the fluid stream). The highest levels of power and efficiency are associated with very large foil pitch angles (upwards of 70°) and angles of attack (30–40°), such that the flow is massively separated for much of the flapping cycle, in contrast to rotary turbines which rely on attached flow over as much of the rotor disk as possible. This leads to leading edge vortices comparable in size to the foil chord, and the evolution and interaction of these vortices with the foil as it moves play a central role in determining performance. The other parameters also influence the vortex behaviour, but in general to a lesser degree. Numerous gaps in the research literature and outstanding issues are highlighted.