The hydrodynamical processes involved in the swimming of real fish are so complicated that analysis is probably only practicable for idealized models. We consider here a drastically idealized case for which a fairly simple mathematical solution can be found. The body is assumed to be very slender, gently tapering, and without any abruptly projecting tins, and to execute motions which are fish-like but of very small amplitude only: the flow over the body is assumed to be effectively the same as that of an inviscid ‘perfect’ fluid. This idealized case has some points of resemblance to swimming snakes and eels, and it is hoped that the analysis may throw some light on the physical nature of the hydrodynamic forces propelling such creatures and on the propulsive efficiencies attained. The first section below discusses propulsive efficiency, and the mathematical solution for the idealized case is presented in the second section. The relevance of this solution to the swimming of animals such as snakes and eels is then discussed.