  An explicit canonical construction of monopole connections on non trivial
U(1) bundles over Riemann surfaces of any genus is given. The class of monopole
solutions depend on the conformal class of the given Riemann surface and a set
of integer weights. The reduction of Seiberg-Witten 4-monopole equations to
Riemann surfaces is performed. It is shown then that the monopole connections
constructed are solutions to these equations.
