The dollar exchange rate with the euro as of Friday is $1.4156. I believe it was $1.19 when the euro officially began and the euro almost immediately declined with respect to the dollar. Eventually, one euro bought less than 90 cents. In recent years, that trend reversed and not long ago, some thought it would hit $1.50.
In today's New York Times, Christina Romer has a column on the exchange rate. In it, she argues that the exchange rate is just a price, a view that most economists would agree with. In general, I do too. However, there are three caveats in my agreement. The first is straight-forward--manipulation of the exchange rate to affect trade patterns should be avoided. Once one country begins it is difficult for others to not follow, and soon we have a kind of beggar-thy-neighbor policy causing problems. Second, when people are uncertain over the state of monetary policy, the exchange rate provides some evidence. The Fed is trying to have an easy money policy right now, and the exchange rate is evidence that they are succeeding even if unemployment remains high.
Finally, all statements about exchange rates being merely prices fail to reflect fully the fact that the dollar is still the reserve currency in the world. Further, the U.S. receives substantial benefits from that status. If other countries, and especially businesspeople and those in financial markets develop a lack of trust in the dollar, the special status of the dollar could switch to another currency. At this point in time, the euro would appear to be the likely candidate, but the Eurozone has problems of its own. China has indicated in the past that it would like a greater role for the reminibi. However, so long as it manipulates its currency, it is unlikely China's currency will replace the dollar. But, forty or fifty years from now, who knows?