Recent situation in era of fast globalization makes immense emphasis on the importance of intercultural interaction between people. Thus, university graduates are required to possess intercultural competence in addition to strong academic skills as well as to overcome communication barriers caused by a clash of different cultures during interactions and encounters. This article discusses the way of communication competence is defined in various theoretical models by intercultural scholars. First, we considered the terms intercultural communicative competence (ICC) and intercultural competence (IC), for which critical cultural awareness is a fundamental element. We also investigate some crucial language barriers, which hinder mutual intercultural communication and implications of this analysis for foreign language education. The concept of language awareness lies in the field of foreign language teaching. The verbal (linguistic) side of intercultural competence has a complex structure and appears in various styles (different styles and genres, slang, colloquial or literary language, dialect, etc.) All these verbal communication characteristics determine the success or failure of communication. Consequently, the so-called “language barriers” of three types (stylistic, semantic and phonetic) were discussed as well. It appears that to overcome language barriers or misunderstandings in field of intercultural communication requires developing the forms of intercultural interaction, communication competence as well as specialized training in this area.