A modern higher education institution is an example of such educational environment where representatives of different nations communicate and collaborate. The success of the mutual activity depends on the awareness of the interlocutors' culture, their own communicative abilities, the previous experience and skills of intercultural communication. This paper attempts to show various aspects of interpersonal communication and interaction of the students with diverse language, cultural and religious background. Some issues of the linguistic and cultural identity of the students, their adaptation and assimilation are examined. Peoples' Friendship University of Russia (RUDN University) is an example of multinational, multicultural environment where young people from many countries study. The common educational environment provides great opportunities for the internalization and socialization of the students and their mutual understanding of the current realities. The practical experience of teaching and training the representatives with diverse language norms and cultural traditions is presented. The findings of the paper are supposed to contribute to further understanding of the ideas of linguistic and cultural Identity of the young people studying at the higher education institutions and to ensure the efficacy of these concepts.