The importance of forming intercultural competences and various aspects of the development of an employee's intellectual activity in today's multicultural society has been stressed. Urgent problems of cross-cultural training of employees, such as the formation of intercultural (cross-cultural) competence of future employees in the process of their professional training, the necessity of compiling a complete list of intercultural competences, determining the levels of requirements for the competences of bachelors and masters, development of cultural intelligence, improvement and the introduction of methods for the development of cultural intelligence have been analyzed. It has been defined that as the most important part of culture, education is intended to ensure the entry of a man into culture through their acquisition of cultural values. The main approaches to the study of cultural diversity in educational systems have been analyzed. The necessity of training specialists with advanced cultural intelligence that will carry out professional activity in the intercultural educational space has been revealed. Four principles, the implementation of which within the framework of corporate culture allows building "the third culture", have been highlighted: unity in diversity; integration; cognitive variety; communicative diversity. Three main competencies that reflect the essence of cross-cultural training have been characterized. They are: awareness of social constructiveness of "cultural differences" and the ability to prevent their influence; skills to identify and evaluate cognitive differences which are hidden behind differences of identities; the ability to find and create a "functional interaction" between cognitive differences of employees, which can contribute to the stability of self-regulating integrative organizational systems.