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In the biggest monetary changeover in history, more than six billion notes and nearly 40 billion coins have been distributed to shops, banks and cash machines.
The French President, Jacques Chirac, described the development as a "new way of being in Europe."
Just three EU countries have refused to part with their currencies: Britain, Sweden and Denmark.
However, European Commission President Romano Prodi said the arrival of the single currency inevitably meant "greater convergence of economic rules" within the eurozone - and any countries outside the zone risked being left out.
Traditional New Year parties became welcome parties for the new common currency in many European capitals.
A million revellers turned out in the German capital, Berlin, and other capitals including Paris, Helsinki and Athens staged special celebrations.
The new currency is unlikely to have its full effect on day-to-day transactions until tomorrow as many businesses are closed for New Year's Day.
National currencies will continue to circulate alongside the new euro for another two months in most member countries.
One notable exception is Germany, where a loophole in the law means the Deutschmark ceased to exist at midnight.
The euro has been used as an electronic currency for financial trading since 1999, but this is the first time it has appeared in the purses of the 300 million ordinary citizens of the eurozone.
The development has intensified the fierce debate in Britain, where the government is committed to a referendum on the issue once certain economic tests set by Chancellor Gordon Brown are met.
Europe Minister Peter Hain defended the government's position, saying, "We want to join but we will only join when it is economically right for Britain to do so."
But, he added, "The increasingly urgent question for British citizens is whether we want to get left behind yet again as we have been consistently in Europe."
The introduction of the euro was widely hailed as a success and went far more smoothly than had been expected.
Within five days, more than half of all cash transactions in the eurozone were being made in the new currency.
The British Government reviewed its decision on the UK's membership of the euro and reported back in June 2003.
The Chancellor, Gordon Brown, said four out of the five tests he set had not been met, and therefore the time was not yet right for Britain to join the currency.
The following year, in his budget statement, Mr Brown again said circumstances had not changed enough to justify re-running the tests.
The "no" campaign against the euro was then wound down as the new focus of debate became the new European constitution.
In September 2003 the Swedish people again voted in a referendum not to join the eurozone.
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