Vortex motion on two–dimensional Riemannian surfaces with constant curvature is formulated. By way of the stereographic projection, the relation and difference between the vortex motion on a sphere (S2) and on a hyperbolic plane (H2) can be clearly analysed. The Hamiltonian formalism is presented for the motion of point vortices on S2 and H2. The set of first integrals for each Hamiltonian shows a corresponding algebraic property in terms of the Poisson bracket defined respectively for S2 andH2. As an example of analytic solutions, the motion of a vortex pair (dipole) is considered. It is shown that a dipole draws a geodesic curve as its trajectory on S2 and H2.