In this article the author considers the issues related to modeling of modern system of senior professionals training of higher level. It focuses on the way specialists are trained in the situation of intensive science and technology development, informatization of living conditions and the process of globalization. It is also stressed by the author, that in modern psycho-pedagogical research the emphasis is primarily laid on the problems of professional competence formations in the process of students training to work in the "person-person" system. These elements of professional competence should be formed in the field of each component of the structure of higher education while their reflection should be traced in their objectives and training programmes, as well as in the requirements to the professional development of lecturers of higher educational establishments, including students. The author characterizes the most important tasks of teaching in higher educational establishment at the current stage of society development, that is knowledge and experience transferance, the use of scientific methods of teaching, the development of students’ values and social relations, the development of management skills which are connected with a sense of responsibility, the formation of readiness to participate in the social and cultural and national and patriotic development of the younger generation, the training of professional staff in various fields. The basic emphasis in the article under consideration is laid on the concept of personal maturity as the most urgent component in the field of professional competence of high school experts. The analysis of various approaches and conceptual positions makes it possible to highlight the main features and characteristics of the psychologically mature person, as well as its basic components and criteria, levels of psychological organization, definite features which serve as important indicators of a mature and immature personalities.