The proposed article analyzes the peculiarities of the basic structural elements of the training systems of teachers of natural and mathematical subjects in postgraduate education of Ukraine and the Republic of Poland. The notion of professional development of specialists in the educational space of the countries is defined — intentional, planned and continuous process, which results in the expansion of teachers’ professional competences, improvement of their professional level, re-qualification of specialists, and development of universal personality. The principles on which the systems of professional development of specialists of Ukraine and Poland are built and developed are considered: humanization, differentiation, interactivity, unity of socialization and professionalization of personality, modularity and continuity of training. Also it is recognized among the principles of educational activities that are taken into account in the process of training of the Polish colleagues there are such as: the interrelation of theory and practice, variability, diversity of forms, continuity, multidimensionality, multilevelness, and openness. Important functions (socio-cultural, developmental, educational, compensatory and adaptive) that the systems of teacher training fulfill as for continuous professional development and improvement of specialists are analyzed. The author pays special attention to the analysis of functioning of the training systems of teachers of natural and mathematical subjects as multidimensional entities in postgraduate pedagogical education of the countries which are characterized by socio-state character, variability, flexibility, and construction considering different needs of teachers in their professional development. The article analyzes the level structures of advanced training systems of postgraduate education of teachers of natural and mathematical subjects of Ukraine and the Republic of Poland, which have their own subjects at each level of regulation (national, regional, local, and institutional), and use certain principles, tools and institutions for achievement of the overall goal. It is noted that the systems of advanced training of teachers of natural and mathematical subjects in the postgraduate education of Ukraine and the Republic of Poland have a lot in common at the stage of formation of goals and tasks, defining principles and functions, choosing the directions of development of educational activity. Differences are observed in the structure of education systems and teacher training financing. A detailed study of the quality criteria for the improvement of specialists in accordance with the international standards is proposed for further research, which will improve the evaluation of the results of the educational process and enhance the motivation of teachers for their personal and professional development and self-improvement