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Character theory of finite groups has an important place in understanding the structure of groups. In the literature, there are many publications from past to present on the relationships between the structure of finite groups and their complex irreducible characters. Sometimes in order to understand the structure of a finite group, instead of working with all irreducible complex characters of it, some special irreducible complex characters of the group have been studied. Let $G$ be a finite group and let $\chi$ be an irreducible complex character of $G$. If $G /\textrm{ker}\chi$ has only one minimal normal subgroup then the irreducible character $\chi$ is called a monolithic character of $G$. In this talk, we will talk about the strongly monolithic characters, which are a subset of the set of monolithic characters of finite groups. In addition to giving some relationships between the structure of finite groups and their strong monolithic character, we will also talk about the classification of finite groups with their strongly monolithic characters. This is a joint work with S.B. Güngör and J.M. Özkan. The work in the talk was supported by (TÜBİTAK). The project number is 119F295.