Following the French and Dutch "no" to the Constitution, the EU Directorate-General Communication commissioned a wide-ranging Special Eurobarometer on the subject of Europe's future. One of the main aims of the survey was to take stock of European people's expectations of different policies. In this article, we analyse EU citizens' viewpoints on the assignment of competence to EU level in matters of social affairs, gender equality and health. Specifically, we focus on country differences and individual characteristics that explain EU citizens' opinions about these issues. Furthermore, we are interested to study EU citizens' views about the harmonisation of social welfare systems within the European Union. This is a particularly interesting question because the harmonisation of social welfare systems is ruled out in EU treaties. For the Special Eurobarometer, almost 25,000 people were interviewed in the 25 EU Member States using the Eurobarometer survey methodology. The survey frame consists of those aged 15 years and over. A multi-stage, random (probability) sample design was applied in all Member States. In each country, a number of households were drawn with a probability proportional to the population size (for a total coverage of the country) and to population density. In each household, a respondent was drawn at random. All interviews were conducted face-to-face in people's homes by using the appropriate national language. We analyse EU citizens' responses to the assignment of competence and harmonisation by using probit and ordered probit models. There is considerable support among EU citizens for the view that more competence should be assigned to the EU level. In most EU member countries, the majority of the respondents favour the idea that more competence should be given to the EU level on the social dimension. However, the results reveal large country differences in notions about the assignment of competence to the EU level in matters of social affairs, gender equality and health. Citizens from Cyprus, Luxembourg and Belgium are most in favour of this particular idea. Approval of the assignment of competence to the EU level is also strong in Portugal. One explanation for this is that citizens in Portugal see the European Union as a vehicle for modernisation. In contrast, citizens from the UK, Estonia and Austria take the most negative stand on this issue. The most important individual characteristics that explain EU citizens' viewpoints about the strengthening of the social dimension are respondent's age and their opinions about the EU and globalisation. In matters of social affairs, gender equality and health, young people are substantially more inclined to assign competence to the EU level than older people. In addition, those who take a positive stand on the EU and/or globalisation are favourable to the idea of giving more competence to the EU level. Based on these results, it seems that EU citizens have accepted the idea that the European Union has a prominent role to play in strengthening the social dimension, because individual EU member countries are too small players in the global economy to wield any influence on these issues. Interestingly, young people take the most critical position on the harmonisation of social welfare systems within the European Union. Furthermore, those who view globalisation positively are also favourable to the idea of harmonising social welfare systems within the European Union. [ABSTRACT FROM AUTHOR]