Abstract: Multiply connected mesoscopic: superconductors are considered within the framework of the nonlinear Ginzburg-Landau theory. The two coupled nonlinear equations are solved numerically and we investigated the properties of a superconducting ring, two concentric rings, and an asymmetric ring. We find that (i) for a mesoscopic superconducting ring the flux through the hole is not quantized, (ii) two concentric mesoscopic superconducting rings are magnetically coupled and the interaction energy increases with increasing sample thickness, and (iii) in asymmetric rings, a stationary phase slip state is predicted.