Euro is becoming the official currency in about fourty countries, in fact. Aside from the EMU member states, this includes the following countries in Europe that were previously using the Deutschmark, lira or Franc:
The list also includes fourteen African nations whose currencies were tied to the French Franc. All of the European (mostly French) dominions/protectorates/etc from the Carribean to the Pacific will be switching to the euro: in fact, the euro's fire christening won't even happen in Europe -- the first people who will be using it are supposed to be the residents of La Reunion, an island in the Indian ocean.
Several other countries, candidates for EU membership, are surely considering making the euro official, however, they probably won't do anything about it without the approval of the ECB.
It is expected that the euro will become the dominant currency of the whole of Europe, and will start to really rival the dollar on the world markets.
Updated 2001-12-31: from the newspapers
Updated 2002-1-3: the journalist was on crack, Estonia's currency is merely linked to the euro