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