Once a vital part of a mighty sea power, Positano is today a sophisticated resort on the central Amalfi Coast. Moorish-style architecture rises up steep slopes that gaze out on the Sirenuse Islands. Smart boutiques, selling fashions for visitors to display on Grand Beach, abound in the village. And it’s a great base for exploring the area—you can easily travel by boat to Capri, Ischia and the Grotta dello Smeraldo cave.

Two thousand years of history have lent Mainz a cosmopolitan air, and museums and cathedrals of various architectural styles dot the city. Bring home a book as a souvenir—Gutenberg worked here, and the museum dedicated to him is a highlight.

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.