Cairo

Cairo, Egypt, has been called "the mother of the world" and "city of victory." Visitors will find it to be a fascinating and often bewildering mixture of old and new. Cairo businesspeople in suits and locals in traditional robes can both be found at sidewalk coffee shops, and minarets and domes share the skyline with high-rise office buildings and hotel towers. Traditional music competes with jazz or Egyptian pop, as well as with the incessant honking of horns.

Cairo's layers of ancient, medieval and modern can be a bit overwhelming. Many things take longer than they should, and nothing works quite perfectly. Patience is a virtue: The expression Ma'alesh (which translates loosely as "Don't worry about it") seems to be on everyone's lips—especially when you're in a hurry. If you set reasonable goals, take frequent breaks and drink plenty of water, you'll have a memorable time in Cairo.           

Dubrovnik

Like something out of a picture-perfect fairy tale, the Old Town of Dubrovnik, Croatia, is a walled medieval city, with drawbridges (used in the 1991-92 war) and 18-ft-/6-m-high gates guarding the main entrances. The entire city is a UNESCO World Heritage site, but it's also very much a living, breathing city.

From the crenulated ramparts and watchtowers, there are some perfect vistas of the city and the Adriatic Sea. Because the Old Town is blissfully free of traffic, the main streets (Placa or Stradun), squares and alleyways are perfect for exploring the city on foot.

Most of the inhabitants of Dubrovnik live outside of the Old Town's city walls. However, inside those walls, the streets and alleyways are crammed with tiny shops, bars, cafes and restaurants that spill out onto the street at every conceivable point.

The serious damage from the Serbia-Croatian War has been completely repaired. Locals are still keen to point out the shrapnel and bullet damage that has been retained as a reminder of those dark days, which still come up frequently in conversation.

Dubrovnik has become one of the most popular tourist destinations in Europe because of its warm climate, proximity to the sea and its rich historic and cultural heritage. In addition, many Game of Thrones fans visit the city to see the original locations that were used in the popular TV show. Its increased popularity also means that the city is facing the challenge of accommodating more visitors, especially during the busy summer season. Therefore, the number of visitors in the old town at any one time is limited to 8,000 people in order to protect historic buildings.           

Lisbon

The hodgepodge of historical periods and cultures represented in Lisbon, Portugal, is a major source of its charm and travel appeal. A sprawling city on the banks of the Tagus River, Lisbon constantly reminds travelers that Portugal has been conquered several times, that it developed (and lost) its own illustrious empire and that, for much of the 20th century, it isolated itself from the rest of the world.

But when Portugal joined the European Union in 1986, it experienced a major economic boost. A completely new quarter sprang up on the banks of the Tagus. Lisbon is modernizing fast as a European travel destination.

As visitors to Portugal walk Lisbon's hills—or, better, take one of Lisbon's vintage trams—they'll find restored medieval facades, wonderful art-nouveau buildings, black-and-white mosaic sidewalks (known as calcada), fine museums and plenty of modern shops.

Lisbon's citizens seem to have absorbed their city's many-sided character. Visitors can witness the popularity of fado, the melancholy music that developed in Lisbon in the early-19th century. Though the performers sing about tragedy and distant glory, the audience is very much a part of modern Lisbon—a flourishing, fashionable business and leisure center.           

Bangkok

Bangkok, Thailand, can soothe or ruffle, and it often does both. While contemplating the sunrise at a temple or monks collecting alms, you'll marvel at what peace can be found in the midst of such a chaotic metropolis.

A cultural hub in Southeast Asia, Bangkok is a collage of urban squalor, gleaming affluence, mass consumerism and pollution. Most certainly, the city will assault your senses. It's fascinating and indulgent, but it requires time and patience.

Comfort of one sort or another is never far away: When your feet tire of wandering through the Grand Palace, head over to neighboring Wat Pho for an hour-long foot massage; if you exhaust your meager supply of Thai words bargaining for souvenirs, pop into an air-conditioned movie theater and take in an English-language film; if you are overwhelmed by the density of people along Sampeng Lane in Chinatown, drop into a coffee shop or open-air restaurant and enjoy a cool drink; if you are looking for excitement, you can watch Muay Thai (kickboxing) at Lumpini Boxing Stadium or hang around with backpackers at the bars on Khao San Road.

Upon arrival, you may find the constant din shocking, the geography impossible (no map does justice to the city's meandering lanes), and the traffic absolutely unbelievable. But stay more than a night or two and the city's bewildering kaleidoscope begins to make sense. Although the cacophony will never melt away, soon it will transmit the excitement and vibrant charm of one of the world's greatest cities.