First, a little history. We've had a Europe-wide currency system before: it was introduced by the Romans in about the year 880, then expanded by Charlemagne, and brought to Britain by William the Conqueror. Throughout Europe we used to have the pound (librium) - the lira in Italy and the livre tournoise in France.
In the 15th century the Italians invented the modern banking system based on double-entry accounting. This led to a rapid growth in bills of exchange. The banks, or rather traders, from different countries would get together at fairs known as assiuntos to settle up. It was one of the world's first clearing systems. But sadly, in spring 1570, when they got together in France to exchange their pieces of paper and settle in real money - gold - they found that all they had was Italian debt! This caused a collapse of the settlements process. The French abandoned the livre tournoise and introduced a new currency, the ecu au soleil. As you can see, we've been here before.
Over the last few hundred years, there have been many periods of floating exchange rates as well as fixed exchange rates.