Abstract: I will give a survey of results around Gap Conjecture which was formulated by speaker at $1989$ and which states that if growth of a finitely generated group is less than $e^{\sqrt n},$ then it is polynomial. If proved, it will be a far reaching improvement of the Gromov's remarkable theorem describing groups of polynomial growth as virtually nilpotent groups. Old and new results around conjecture and its weaker and stronger forms will be formulated and discussed

This talk will be accessible to graduate students.