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| Credit: Nick Hanna / Alamy

It was in the mazelike bazaars of Kalbadevi that I encountered the American. Veiled by smoke from street-stall frying pans, trailed by an enormous cow, he swam toward me through the crowd: drenched in sweat, wide-eyed, and lost. Bombay will do that to you.

"Maybe you can help me," he said, catching his breath. "I'm trying to get back to the tourist area?" A strange request, considering that this marketplace is a tourist area, inasmuch as Bombay has one. But he had something less frenetic in mind—perhaps with air-conditioning. I pointed the way out, and off he retreated to his hotel. Another shell-shocked initiate on his first tour of duty.

The crowds and chaos of Bombay can bewilder even the savviest of travelers (by 2015, this is projected to be the world's most populous city), but settle into its rhythms and you'll soon be won over. Bombay is India at its most contradictory: aggressively modern, yet in parts verging on medieval; glamorous, yet rough-edged; dazzlingly cosmopolitan, yet quintessentially Indian. I've traveled from Goa to the Himalayas, and I've never felt the same romantic charge that I get from Bombay. I try to return as often as I do to Paris or London, and with each visit, it seems more like both: a well that grows deeper as you draw from it.

There are many things to love about Bombay. One is simply the geographical names, like The Queen's Necklace and Elephanta Island, all with roots in the British Empire. Bombay itself has disputable origins. Is it a variation on the goddess Mumba?Or an anglicization of the Portuguese buan bahia ("good bay")?Whatever the answer, Hindu fundamentalists cast off the colonial moniker in 1996 and renamed the city Mumbai. This struck some as a spectacularly bad marketing move ("as if McDonald's had renamed itself Kroc's in honor of its inventor," wrote the author Shashi Tharoor). In any case, most English-speaking Indians still use Bombay.

Incidentally, the next time I saw the American—a week later, strolling along the waterfront—he had the beatific look of a convert. When I caught his eye, he flashed a grin that said, Now I get it.

"Bombay is not about 'sights' in the traditional sense," says my friend Rashida Anees, a tour director and lifelong resident. "It's about experiences."