How can big data help to understand the migration phenomenon? In this paper, we try to answer this question through an analysis of various phases of migration, comparing traditional and novel data sources and models at each phase. We concentrate on three phases of migration, at each phase describing the state of the art and recent developments and ideas. The first phase includes the journey, and we study migration flows and stocks, providing examples where big data can have an impact. The second phase discusses the stay, i.e. migrant integration in the destination country. We explore various data sets and models that can be used to quantify and understand migrant integration, with the final aim of providing the basis for the construction of a novel multi-level integration index. The last phase is related to the effects of migration on the source countries and the return of migrants.