The nonlinear saturation mechanism based on the magnetic helicity evolution is discussed. It is shown that this universal mechanism is nearly independent of the form of the flux of magnetic helicity, and it requires only a nonzero flux of magnetic helicity. Different forms of the flux of magnetic helicity are discussed. We also studied a simple model for the solar dynamo in the framework of the Parker migratory dynamo, with a nonlinear dynamo saturation mechanism based on magnetic helicity conservation arguments. We found a parameter range in which the model demonstrates a cyclic behaviour with properties similar to that of Parker dynamo with the simplest form of algebraic alpha-quenching. We compared the nonlinear current helicity evolution in this model with data for the current helicity evolution obtained during 10 years of observations at the Huairou Solar Station of China. On one hand, our simulated data demonstrate behaviour comparable with the observed phenomenology, provided that a suitable set of governing dynamo parameters is chosen. On the other hand, the observational data are shown to be rich enough to reject some other sets of governing parameters. We conclude that, in spite of the very preliminary state of the observations and the crude nature of the model, the idea of using observational data to constrain our ideas concerning magnetic field generation in the framework of the solar dynamo appears promising.