The article deals with the problems of training scientific and pedagogical personnel in Russia at the present stage of postgraduate education, which caused the need to discuss them, both at the level of the country’s leadership, and at the level of the scientific and University community. The directions of improving the organization of modern postgraduate studies in the context of a systematic approach are proposed. The directions of improving the training of postgraduates and the organization of the modern Russian postgraduate school in the context of its two types - scientific and University with the organization of different types of practice: research and teaching. The most significant among the directions proposed in the article are the following: improving the quality of recruitment of applicants to graduate school on the example of a pre-admission system of two-stage training of applicants in the system of professional development of the Research center for problems of quality of training of specialists; the implementation of master’s training programme “graduate student researcher” and “student-expert” on the basis of the principle of continuity of training programs for “masters-postgraduate” and implementation of the program and the model of problem-designresearch teaching academician I.A. Zimnyaya; formation of academic literacy in the process of formation of communicative competence of graduate students by means of educational disciplines «Academic writing» / «Scientific text», «Academic speech»; improving the organization of classroom and extracurricular independent work of postgraduates and its material and technical support; financing of postgraduate education; and a number of others. Along with the actualization of a systematic approach to improving the organization of modern graduate school activities, personal-activity, convergent and scenario approaches to improving the training of graduate students are identified. The proposed ways to solve the existing problems of the organizational plan of the postgraduate school in some of the indicated areas can have a positive impact on its improvement.