We consider observations and data from live fish and cetaceans, as well as data from engineered flapping foils and fishlike robots, and compare them against fluid mechanics based scaling laws. These laws have been derived on theoretical/numerical/experimental grounds to optimize the power needed for propulsion, or the energy needed for turning and fast starting. The rhythmic, oscillatory motion of fish requires an “impedance matching” between the dynamics of the actively controlled musculature and the fluid loads, to arrive at an optimal motion of the fish’s body. Hence, the degree to which data from live fish, optimized robots, and experimental apparatus are in accordance with, or deviate from these flow-based laws, allows one to assess limitations on performance due to control and sensing choices, and material and structural limitations. This review focuses primarily on numerical and experimental studies of steadily flapping foils for propulsion; three-dimensional effects in flapping foils; multiple foils and foils interacting with bodies; maneuvering and fast-starting foils; the interaction of foils with oncoming, externally-generated vorticity; the influence of Reynolds number on foil performance; scaling effects of flexing stiffness of foils; and scaling laws in fishlike swimming. This review article cites 117 references.