This talk will be devoted to the long study of actions of groups on the circle. The main conjecture is the following: if a smooth action of a finitely generated group on the circle is minimal then it is ergodic. This conjecture is still an open problem but many partial results were obtained is special cases as well as new class of actions (with so-called non-expandable points) were studied. I will speak about the progress in this project up to the latest results.