Laid out by British architect Edwin Lutyens, the Indian capital is a striking modern metropolis. A gracious contrast to Old Delhi's winding streets, the grand avenues and stately buildings of New Delhi are rich with history and culture, from Gandhi's Delhi home (and the site of his assassination) to the tomb of Humayun, a complex of Mughal buildings reminiscent of the Taj Mahal. Chaotic traffic is best left to the locals. Negotiate a good price for taxis or travel on the new Delhi Metro.

Udaipur, known as the Venice of the East, boasts several sparkling lakes against a backdrop of the Aravail hills. Jag Niwas and Jag Mandir, islands in Fateh Sagar Lake, are the site of Udaipur Solar Observatory and Nehru Garden. Famous palaces include the magical Lake Palace, now a luxurious five-star hotel, and the massive City Palace on Pichola’s east bank, featuring epic courtyards and stunning paintings.

Known as both the "Garden City" and "The Silicon Valley of India," Bangalore (officially "Bengaluru") is a techie’s paradise, boasting the highest concentration of IT companies in the country. When you’re done geeking out, there are plenty of gardens, museums, natural features, palaces and temples to fill your dance card. Visit Vidhana Soudha, Cubbon Park and the Ulsoor Lake of Bangalore, well known for its beautiful locales and boating facilities. Bangalore is also a major centre of Indian classical music and dance, and of vivid, cutting-edge nightlife.

Vietnam's bustling largest city sets the cultural and economic pace for the country. The former Saigon boasts charming French colonial architecture and wide boulevards, usually thronged and choked with traffic. Taxis are an option for seeing the sprawling city. The War Remnants Museum shows the Vietnam War through Vietnamese eyes. Don't miss the impressive Jade Emperor Pagoda. Go to the frenetic Ben Thanh Market for food, flowers or frogs. Tour through the Mekong Delta, past rice paddies and houseboats.

Singapore is known as a bustling metropolis that also happens to be one of the cleanest and safest cities of its size in the world. (Just make sure you heed the local laws—something like spitting in the street might merely be considered rude in your hometown, but here, it carries a severe penalty.) You'll find historic sites like the Thian Hock Keng temple, superlative shopping (including gargantuan malls) and numerous beaches.

Shanghai is the cool, confident face of modern China, and its energy is infectious. Go to the Bund to watch ships on the river and marvel at the huge variety of architectural styles on display, or watch the crowds go by in People’s Square. Shoppers should make a beeline for the Fabric Market, where you can have a suit or dress tailor-made for you at bargain prices. At night, explore all manner of fashionable restaurants, bars and nightclubs or just stroll through the city enjoying the spectacular neon lights.

Over 15 million gallons of water bubble daily into Budapest's 118 springs and boreholes. The city of spas offers an astounding array of baths, from the sparkling Gellert Baths to the vast 1913 neo-baroque Szechenyi Spa to Rudas Spa, a dramatic 16th-century Turkish pool with original Ottoman architecture. The "Queen of the Danube" is also steeped in history, culture and natural beauty. Get your camera ready for the Roman ruins of the Aquincum Museum, Heroes' Square and Statue Park, and the 300-foot dome of St. Stephen's Basilica.

If you currently think your neighbourhood coffee shop is nice, you might want to stay out of Vienna’s coffeehouses. After you’ve gotten used to these palatial, yet welcoming cafes—and their delicious coffee and Sacher torte—your local café will pale in comparison. Between coffee breaks, visitors can explore Vienna’s Schonbrunn Palace and Imperial Palace. And if you have a chance, catch a performance at the State Opera House—it’s not to be missed.