The internationalization and specialization of science confront scientists with opportunities and sometimes even a need to become internationally mobile during their careers. Against this background, we systematically reviewed empirical studies on the effects that mobility across national borders has on the careers of scientists. Using several search channels, we identified 96 studies -- published between 1994 and 2019 -- that examine how international mobility influences eight dimensions of scientists' careers. Listed in descending order of the number of identified studies, these dimensions comprise scientists' (1) international networks, (2) scientific productivity, (3) occupational situation, (4) scientific impact, (5) competences and personality, (6) scientific knowledge, (7) access to research infrastructures and funds, and (8) symbolic capital. Existing research provides robust evidence of positive effects of international mobility on the broadening of scientists' networks. Moreover, several solid studies examine the effect of international mobility on scientists' productivity, impact, and occupational situation. Most of them find positive effects, but some also find no or negative effects. Studies on the other career dimensions are not only less frequent, but mostly also less robust. Our review reveals potential to advance research in the field by using less selective samples and more rigorous methodological approaches. Intending to spur further theory-driven empirical research, we develop a model integrating research on the identified career dimensions and derive various questions for future research. We conclude by highlighting policy implications of existing research.