Many businesses used this period to gain control of domains relevant to brands and products they produce.
During those initial four months, more than 300,000 .eu domains were registered.
But this was dwarfed by the rush of registrations seen during the first four hours that applications were open to all in the 25-nation EU trading bloc.
In total, there were 702,684 applications on the first morning. The first five names to be registered were dekoration.eu, buchung.eu, tankstellen.eu, exhibitionist.eu and eurorechner.eu.
At the peak of registrations, about 76 applications were being processed every second.
"We are satisfied to see that the interest in .eu has been so large and that many already regard .eu as a useful complement to internet's national top level domain names," said Eurid managing director Marc Van Wesemael, in a statement.
Heavy demand
The rush meant that the servers logging the applications and who owns which name were regularly unavailable.
Statistics provided by Eurid show that Germans now own 419,866 of the 1,318,171 active .eu domains. Second in the list of top registrants was the UK with 270,292.
The interest was perhaps high because so many people were trying to claim domain names that are likely to be homes for popular websites.
In a bid to speed up the resolution of disputes over domains, Eurid has set up a virtual adjudication chamber that has a physical base in Prague but will conduct hearings online.
On Friday Eurid reported that the virtual court had got 60 requests to sort out disputes.
