Fish, or in general any mechanical object moving in an inviscid fluid, can be described by means of a number of interesting differential-geometric structures, amongst other bundles and connections, groups of diffeomorphisms, and symplectic reduction. I will describe some of these structures and outline their role in fluid dynamics. Along the way, a number of parallels will appear with other dynamical systems: time permitting, we will describe a Kaluza-Klein description of fluid-structure interactions (making the link with magnetic particles), and we will see how the flux homomorphism from symplectic geometry makes an appearance through an old construction of Kelvin.