It’s nicknamed the Eternal City for a reason. In Rome, you can drink from a street fountain fed by an ancient aqueduct. Or see the same profile on a statue in the Capitoline Museum and the guy making your cappuccino. (Which, of course, you know never to order after 11 am.) Rome is also a city of contrasts—what other place on earth could be home to both the Vatican and La Dolce Vita?

One of Poland’s most beautiful cities, Gdansk, on the Baltic Sea, has played major roles in history, especially in the 20th-century. It was the 1939 flash point of World War II, and then in 1980, the birthplace of the Solidarnosc labor movement, ushering the end of Communist domination in Eastern Europe. Gdansk’s Old Town, painstakingly reconstructed to its Hanseatic League glory after being leveled in World War II, is a highlight. The 14th-century Town Hall houses the city’s historical museum.

There are 2,000 years of history in Cologne, and visitors here will find everything from Roman towers to Gothic churches to fine examples of modern architecture. Cologne has a variety of museums, too—check out the Museum of Applied Art, the Museum Ludwig and, if you have a sweet tooth, the Chocolate Museum. Be forewarned, though—the gift shop at the latter will utterly ruin your diet.

The first time you go to New York, go ahead and be a sight-seer—everyone should visit the Statue of Liberty, the Met, Times Square, etc. But on a return trip, pick a neighbourhood and go deep. You’ll find hole-in-the-wall bars, great delis, quirky shops… exploring the non-touristy side of New York is an incredibly rewarding experience for a traveller.