In the mid-18th century,Euler derived his famous equations of motion of an incompressible fluid, one ofthe most studied equations in hydrodynamics. More than 200 years later, in1966, Arnold observed that they are, in fact, geodesic equations on the(infinite dimensional) Lie group of volume-preserving diffeomorphisms of amanifold, endowed with a certain right-invariant Riemannian metric. Since then,many important hydrodynamical PDEs were recast, similarly, as geodesicequations, and similar geodesic equations, in turn, turned out to be importantin imaging problems (also known as "shape analysis"). In the talk Iwill present the basic settings of Riemannian geometry on diffeomorphism groups(and similar spaces), and discuss some their geometric properties (e.g.,completeness, diameter). I will then show how this interpretation can help usstudying certain PDEs (like incompressible Euler), and discuss someapplications to shape analysis.