This paper sheds light on the global migration patterns of the past 40 years, and produces migration projections for the 21st century, for two skill groups, and for all relevant pairs of countries. To do this, we build a simple model of the world economy, and we parameterize it to match the economic and socio-demographic characteristics of the world in the year 2010. We conduct a backcasting exercise which demonstrates that our model fits the past trends in international migration very well, and that historical trends were mostly governed by demographic changes. We then describe a set of migration projections for the 21st century. In line with backcasts, our world migration prospects and emigration rates from developing countries are mainly governed by socio-demographic changes: they are virtually insensitive to the technological environment. As far as OECD countries are concerned, we predict a highly robust increase in immigration pressures in general (from 12 in 2010 to 17-19% in 2050 and 25-28% in 2100), and in European immigration in particular (from 15% in 2010 to 23-25% in 2050 and 36-39% in 2100). Using development policies to curb these pressures requires triggering unprecedented economic takeoffs in migrants countries of origin. Increasing migration is therefore a likely phenomenon for the 21st century, and this raises societal and political challenges for most industrialized countries.