  We set up general equations of motion for point vortex systems on closed
Riemannian surfaces, allowing for the case that the sum of vorticities is not
zero and there hence must be counter-vorticity present. The dynamics of global
circulations which is coupled to the dynamics of the vortices is carefully
taken into account.
  Much emphasis is put to the study of vortex pairs, having the Kimura
conjecture in focus. This says that vortex pairs move, in the dipole limit,
along geodesic curves, and proofs for it have previously been given by
S.~Boatto and J.~Koiller by using Gaussian geodesic coordinates. In the present
paper we reach the same conclusion by following a slightly different route,
leading directly to the geodesic equation with a reparametrized time variable.
  In a final section we explain how vortex motion in planar domains can be seen
as a special case of vortex motion on closed surfaces, and in two appendices we
give some necessary background on affine and projective connections.
