A nonlinear thin-disk galactic dynamo model based on α-quenching is proposed. Assuming that the mean helicity depends on the magnetic field strength averaged across the disk, we derive a universal form of nonlinearity in the radial dynamo equation. We discuss the evolution of the regular magnetic field in the Milky Way and the Andromeda Nebula. It is argued that the reversals of the regular magnetic field in the Galaxy are a relic inherited from the structure of the seed field. We also briefly discuss the role of the turbulent diamagnetism and the effects of galactic evolution on the dynamo.