Circular proofs for non-classical logics

Usually, in a logical proof a cycle, that is, using the statement we wish to prove as an argument towards proving it, is considered incorrect (circulus vitiosus). There is, however, a vividly developing field of study of circular proofs, where such cycles are allowed to be used, under certain conditions, without losing logical validity. In this talk, we survey several applications of this approach in non-classical logics, namely, modal logics and substructural logics with Kleene star.