What kind of Europe do Europeans want?

After two crisis decades, the European Union (EU) needs to meet the challenges of climate change, migration, strategic competition, and war. But, given these challenges, what kind of Europe do Europeans want? The article argues that the idea of more or less integration does not adequately capture different polity preferences since more integration can mean different things to different people. In order to capture this polysemy, the article presents a richer conceptualisation of European integration along two dimensions: binding (EU centralisation of competences) and bounding (bordering at the EU level). Combining these two, five types of polity formation preferences are constructed: Fortress, Sovereign, Cosmopolitan, Minimalist, and No-vision EU. Exploiting a new original survey, the analysis confirms that public preferences are well represented by these two dimensions. A majority wants more bounding, but they are divided on binding.