We study the impact of foreign direct investment (FDI) on total factor productivity (TFP) of domestic firms using a new, representativ firm-level data set spanning six countries. A novel finding is that firm-level spillovers from foreign firms to domestic companies can be significantly positive, non-existent, or even negative, depending on which sectors receive FDI. When foreign firms produce in the same narrow sector as domestic firms, the latter are negatively affected by increasing competition and positively aected by knowledge spillovers. We find that the positive spillovers dominate if foreign firms enter sectors where firms are "technologically close," controlling for the endogeneity of their entry decision into such sectors. Positive technology spillovers also affect firms in other sectors, if those sectors are technologically close to the sectors receiving FDI. Increasing FDI in sectors that are technologically close to other sectors boosts TFP of domestic firms by twice as much as increasing FDI by the same amount across all sectors.