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Abstract: The Euler equations of hydrodynamics are used to describe fluid flows in space. An interesting special case are the so-called axisymmetric flows, which serve as a model for phenomena such as hurricanes or tornadoes -- flows that spin around some axis, and therefore carry some symmetry. In order to gain some insight into difficult questions about these flows, we generalize the notion of axisymmetry to any Riemannian 3-manifold M carrying a Killing field, and describe how the equations change according to the geometry of M. We then prove that the data-to-solution map associated with axisymmetric flows is Fredholm, which was not known even in the classical case of R^3, and discuss some applications. This is joint work with G. Misiolek and S. Preston