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Let $\Sigma$ be a compact manifold without boundary whose first homology isnontrivial. Hodge decomposition of the incompressible Euler's equation in termsof 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components area `pure' vorticity flow and a potential flow (harmonic, with the dimension ofthe homology). In this paper we focus on $N$ point vortices on a compactRiemann surface without boundary of genus $g$, with a metric chosen in theconformal class.  The phase space has finite dimension $2N+ 2g$. We compute a surface ofsection for the motion of a single vortex ($N=1$) on a torus ($g=1$) with anon-flat metric, that shows typical features of non-integrable 2-dofHamiltonians. In contradistinction, for flat tori the harmonic part isconstant. Next, we turn to hyperbolic surfaces ($ g \geq 2$), having constantcurvature -1, with discrete symmetries. Fixed points of involutions yieldvortex crystals in the Poincar\'e disk. Finally we consider multiply connectedplanar domains. The image method due to Green and Thomson is viewed in theSchottky double. The Kirchhoff-Routh hamiltonian given in C.C. Lin's celebratedtheorem is recovered by Marsden-Weinstein reduction from $2N+2g$ to $2N$.  The relation between the electrostatic Green function and the hydrodynamicalGreen function is clarified.  A number of questions are suggested.

On the interplay between vortices and harmonic flows: Hodge decomposition of Euler's equations in 2d