What happens with an individual's values when he or she migrates to a new country with other social norms than in the country of origin? In this project, we study cultural change among migrants by analyzing individual-level survey data from a large sample of non-European migrants in Sweden collected through the Migrant World Values Survey. The case of Sweden is interesting both for being a country that has recently experienced a large increase in its foreign-born population share, and also for having cultural values quite distinct from the average among its recent immigrant waves. In the study, we first analyze how selected the Swedish immigrants are by comparing the values of newly arrived migrants to the average in their origin country or cultural zone. Second, by comparing similar immigrants that have lived differently long in Sweden, we study the speed of cultural convergence in different cultural dimensions including values related to trust, choice, family, religion, and politics. Third, we also analyze the potential determinants of this cultural integration speed, including gender, age, income, education, employment, language, demographics, origin and cohort effects. Our study contributes to the literature on migration and cultural change, and will have important policy implications regarding the inclusion and integration of migrants.