Exponentially small separatrix splitting

Summary form only given. The history of obtaining and subsequent strict justification of the asymptotic formula for the exponentially small separatrix splitting for the standard map is expounded. These two events are separated by the time interval equal to 15 years. The author describes the main mathematical difficulties and the ways of their overcoming. The reader can find a more detailed exposition of the material of the talk and its generalizations in a survey by V.G. Gelfreich and the author which is in preparation