The first decade of the century was marked by increased migration inflows into numerous European countries. The ability of immigrants to integrate, as well as the ability of receiving countries to promote the equality of chances, has been at heart of vivid debates. However, in general, we lack statistical indicators that would allow understanding the degree of assimilation of immigrants. In this Letter, we compute the indices of similarity between immigrants and native-born for a number of European destination countries, distinguishing between origin countries and several generations of immigrants. Obtained measures suggest a multi-dimensional character of the assimilation processes, which vary along economic, cultural, and civic characteristics of individuals.