This article is concerned with some of the implications of the fact that Europe is so widely seen as a place replete with heritage, museums and memory, and also with the continuing expansion in numbers and types of heritage, museums and memory. It seeks to explore some of the ways in which heritage, in particular, is understood (including what it calls 'sticky heritage'), and especially the cultural and social work that it is often seen as able to do. To this end, the article reviews a number of trends in heritage developments, especially the diversification of what it calls 'Museum Europe' (e.g. in the establishment of museums or exhibitions about migration) and the kinds of citizenship that this mobilises. Some of the dilemmas as well as capacities of these developments are discussed. At the same time, the article reviews some of the directions in heritage research and the implications of this, and of 'Museum Europe' itself, for anthropology, ethnology and related disciplines.