The EU-appointed registrar for .eu domains is EURid, an amalgamation of a Swedish, a Belgian and an Italian registrar.
The .eu TLD (top level domain) is seen as a counterpart to .com, and EURid estimates one million registrations within the first year, private individuals are welcome too.
Prices are not known yet. Interestingly the EURid web pages are completely un-European just in English. However, you have six months left to decide whether you really need a .eu domain. Registrations start then.