The rules behind the .eu top level domain – specifically Regulation 733/2002 – state that only persons, companies or organisations based in the European Economic Area (the EU plus Iceland, Norway and Liechtenstein) can register a .eu domain name.
This would suggest, as the European Scrutiny Committee certainly did in January 2017, that “By default, when the UK leaves the European Union, and any transition period ends, UK persons and organisations that have registered .eu domain names will no longer be legally eligible for these registrations. 340,000 UK users potentially stand to be affected by this development.”
DCMS’s explanatory memorandum on the domain’s status to the European Scrutiny Committee merely quotes EURID, the .eu domain authority, as saying that, “as the next steps have still not been determined and the political and legal processes have not yet been initiated, note that no action will be taken against .eu or .ею domain names that have been registered by residents in UK…When further details are known about the timing and details of a UK exit, the European Commission will instruct EURid on how to proceed. We will continue to keep all our stakeholders fully informed.”
On 28 March the EC confirmed that “as of the withdrawal date, undertakings and organisations that are established in the United Kingdom but not in the EU, and natural persons who reside in the United Kingdom will no longer be eligible to register .eu domain names or, if they are .eu registrants, to renew .eu domain names registered before the withdrawal date.”