In the talk we discuss several ramifications of Arnold’s group-theoretic approach to ideal hydrodynamics as the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms. We show that problems of optimal mass transport are in a sense dual to the Euler hydrodynamics. Moreover, many equations of mathematical physics, such as the motion of vortex sheets or fluids with moving boundary, have Lie groupoid, rather than Lie group, symmetries. This is a joint work with Anton Izosimov.