We analyze the decisions of foreign-born PhD and postdoctoral trainees in four natural science fields to come to the United States vs. go to another country for training. Data are drawn from the GlobSci survey of research scientists in 16 countries. A major reason individuals report coming to train in the United States is the prestige of its programs and/or career prospects; perceived lifestyle in the United States is a major factor individuals report for training elsewhere. The availability of exchange pro- grams elsewhere is associated with fewer PhD students coming to the United States. The relative unattractiveness of fringe benefits in the United States is associated with going elsewhere for post- doctoral training. Countries that have been nibbling at the US PhD and postdoc share are Australia, Germany, and Switzerland; France and Great Britain have gained appeal in attracting postdocs, but not in attracting PhD students. Canada has made gains in neither.