Recent progress in inflationary cosmology

I briefly describe the evolution of the inflationary theory from the scenario based on the idea of supercooling and expansion in the false vacuum toward the theory of eternally expanding self-reproducing inflationary universe. I describe also the recent development of inflationary cosmology with Omega not equal 1, and then discuss some issues related to the possibility to verify inflation by comparing its predictions with observational data. I argue that it is possible to verify and disprove many particular models of inflationary cosmology, but it is very difficult to kill the basic idea of inflation. It seems that the best way to do it is to suggest a better cosmological theory.