Planar superconductors, emerging in thin films with thickness comparable to the superconducting coherence length, differ crucially from their bulk counterparts. Coulomb interactions between charges are logarithmic up to distances comparable to typical sample sizes and the Anderson-Higgs mechanism is ineffective to screen the infrared divergences of the resulting (2+1)-dimensional QED because the Pearl length screening the vortex interactions is also typically larger than the sample size. As a result, the system decomposes into superconducting droplets with the typical size of order of superconducting coherence length. We show that two possible phases of the film match the two known mechanisms for curing the (2+1)-dimensional QED infrared divergences, either by generating a mixed topological Chern-Simons mass or by magnetic monopole instantons. The former mechanism works in superconductors, the latter one governs mirror-dual superinsulators. Planar superconductors are thus described by a topological Chern-Simons gauge (TCSG) theory that replaces the Ginzburg-Landau model in two dimensions. In the TCSG model, the Higgs field is absent. Accordingly, in planar superconductors Abrikosov vortices do not form, and only Josephson vortices without normal core do exist.