I will focus on metrics of Sobolev type. As pointed out by V. Arnold, motions of an ideal fluid in a compact manifold M correspond to geodesics of a right-invariant L^2 metric on the group of volume-preserving diffeomorphisms of M. I will discuss recent results on the structure of singularities of the associated exponential map. Time permitting I will also describe the geometry of an H^1 metric on the space of densities on M and its relation to geometric statistics.