The phenomenon of scientific mobility, actively developing in recent decades, attracts increasing attention of researchers in view of its importance for the development of science, dissemination of scientific knowledge, making informed decisions in the management of science and training of qualified personnel. Based on an extensive analysis of the literature on the topic in the last 30 years with the use of bibliometric approaches, this paper outlines the main evolutionary stages of scientific mobility in the context of brain drain and circulation concepts;​ considers relations, advantages and disadvantages of scientific mobility in relation to scientific inbreeding;​ describes the main approaches and methodological aspects formed today in the study of the scientists mobility;​ discusses its positive and negative consequences for researchers, organizations, countries, and individual disciplines, and summarizes the motivations and driving forces of scientists when leaving the country and when returning.