We develop a model that accounts for several stylized facts about immigrant earnings. First, it implies that new immigrants learn less than natives with the same level of schooling. Second, depending on the age of immigration, it implies that initially, immigrant earnings need not grow faster than natives but that, after some time, measured earnings of immigrants will outpace those of comparable ---using educational attainment--- natives. Finally, the model also implies that the rates of catch up and the earnings gap between immigrants and natives, depend on the individual's country of origin. We develop a calibrated version of the model and we show that it is broadly consistent with the evidence.