In my day, travelling in Europe meant changing your money into French francs, Italian lire, Dutch guilders, Belgian francs, German marks, and Spanish potatoes. You wouldn't necessarily obtain all these currencies before leaving home, you'd get them as you went along. But now, armed with a fistful of euros, you can proceed to the Continent and not worry about borders. Since we arrived in Paris on 6 July, we've crossed an international border nine times, not counting Turkey, and we've only used the euro. Turkey uses 'lire', but then it's arguably not in Europe. But even there, around the port city of Kusadasi, businesses will happily just take euros. Of course, on the Continent, the perennial oddball is Switzerland, still splendidly isolated within its Alps, still happily going its own way, and still using the Swiss franc.
I should say this experience of using a single currency across multiple borders is not my first. When I travelled across Africa in the mid-80s, there were several countries using the 'Central African franc'. I recall using it in countries like Niger, Upper Volta (later Burkina Faso), Togo, Benin, Cameroon and the Central African Republic. You had the feeling that there was more economic confidence in these states than among their neighbours. These were all former French colonies, and, as I recall, the Central African franc was guaranteed by the French treasury.
Europe has been using the euro for only about 15 years. In the last years of the 1990s, it was available in the form of loans or for trading, but not in physical form. From 1 January 2002, you could carry euros in your pocket for the first time, and shop at the corner store. Apart from the BAFFLING PIGS countries (take each letter and work out the country), other EU states using the euro include Slovakia, Slovenia, Cyprus, Estonia and Malta.
European countries using the euro that are not in the EU include Vatican City, Monaco, San Marino and Andorra. Kosovo and Montenegro use the euro as a de facto domestic currency. They have no formal arrangements with the EU.