The U.A.E. customs agent flipped through my passport, found a blank page and flashed me a grin. “Welcome to Vegas,” he said, and slid the passport across the countertop like a croupier dealing cards.

Dubai isn’t exactly Sin City — no casinos, no showgirls, no drinking in the street (or anywhere outside of the sanctioned hotels, for that matter). But it’s an apt comparison nonetheless, and not least for the shared theme of excess-as-entertainment, with ambitious architecture, endless expansion and glitz on a gargantuan scale.

In Las Vegas, that’s mostly confined to a flashy, four-mile stretch known as “the Strip.” But in the desert fantasyland that is Dubai, the show begins the second you step off the plane and into the wing-shaped Terminal 3 of Dubai International Airport.

At 370 acres and more than half a mile from end to end, the $4.5 billion building is the world’s largest by area, with palm trees lining the concourse, the 5-star Airport Hotel and a bustling duty-free zone — also the world’s largest — full of luxury designer brands, blended Scotch whiskeys and entire aisles devoted to Tang.

It’s at once a symbol of the emirate’s outsized aspirations and a means of achieving them. Sheikh Mohammed bin Rashid Al Maktoum, Dubai’s ruler, says he wants to lead an Arab renaissance capable of transforming the region. And while many visitors to the emirate see Vegas, he has his sights set on something far grander: 10th-century Córdoba, Europe’s Arab-ruled center of learning and culture and the home of what was, at the time, the largest library in the world.

While petrodollars paved Sheikh Zayed Road, the city’s mammoth main artery, Dubai’s rulers knew their oil wouldn’t last long. Which is why they decided some years ago to diversify — and to build a city so surreal, so sci-fi, so outrageous at every turn that, in time, it would surprise no one to hear of a black-diamond ski slope inside a supersized shopping mall, or an underwater restaurant accessed by a submarine elevator, or a manmade island shaped like a palm tree, or three of them.

The sheikhs wagered that even the most hard-boiled traveler would at some point succumb to curiosity, that over the long haul tourism would carry the emirate’s economy — and if recent history is any guide, they were right. Despite a global recession and plummeting real estate values, the tourism sector has thrived; the emirate welcomed some 9.5 million visitors in 2010. Of those, close to half a million were Americans, an 11 percent increase over 2009.

I made my first trip to the city last February. Having been in a rainy Beirut for the previous three weeks, I was desperate for some sun. And I knew Dubai would deliver. I also knew that the final round of the Dubai Duty Free Tennis Championships was to kick off that weekend, with Roger Federer taking the court against Novak Djokovic at the Dubai Tennis Stadium.

Housed in the luxurious Aviation Club, a sprawling leisure facility offering everything from squash lessons and salsa classes to fine dining and an Irish pub, the stadium has hosted dozens of the world’s biggest stars since it opened in 1994 — and not only athletes. Sting, Shaggy, the Gipsy Kings and Gloria Gaynor are just a few of the acts to have graced the stage in what’s also one of Dubai’s leading concert venues.

Moreover, it’s just a fraction of what’s in store for spectators down the road. The debt crisis at Dubai World, the state-run investment giant, sent investors fleeing when it hit in 2009, and some of the city’s more whimsical projects — replicas of the Taj Mahal, the Eiffel Tower, and the Great Wall of China, among others — were put on hold or scrapped altogether.

But Dubai’s brush with disaster hardly halted its progress. Even as markets tumbled around the world, construction continued apace on Terminal 3’s desperately needed new concourse; a vast network of highways; and a state-of-the-art subway system, the $4.2 billion Dubai Metro. Now, with its financial woes fading into the past, thanks in large part to a bailout by Abu Dhabi, work is soon to resume on the sort of showstoppers that have made Dubai what it is.

One of those is the massive new Dubai Sports City. Part of the far larger Dubailand development, it’s envisioned as being “a destination for the sporting world”: separate stadiums for soccer, cricket and field hockey; an indoor arena for ice hockey and other hard-court sports; and an 18-hole “desert links” championship golf course, the Els Club, designed by former world’s No. 1 golfer Ernie Els (aka “The Big Easy”).

Of course, the Els isn’t the only course in town. The Emirates Golf Club, host of the Omega Dubai Desert Classic and the Omega Dubai Ladies Masters, features its own pair of championship courses — one designed by golf legend Nick Faldo and the other, the Majilis, consistently ranked among the top 100 courses in the world by Golf Digest. There’s also the Dubai Creek Golf Club with its waterfront fairways and a floodlit par-3 course guests can play after sunset.

Watching that sunset from atop the world’s tallest structure is one of those thrills you can’t put a price on. Unless, of course, you factor in the $1.5 billion it cost to build the sky-high Burj Khalifa. At 2,717 feet, the shimmering, glass-clad tower is more than twice the height of the Empire State Building, and its 124th floor observation deck — aptly dubbed At the Top — affords 360-degree views of the city, including the Downtown Dubai development that surrounds it.

DIVERSIONS

The biggest, the highest, the finest, the first: The story of Dubai is a story of superlatives, and to see it at street level is to glimpse the future — at least one wildly imagined version of it. From the 7-star, sail-shaped Burj Al Arab, consistently ranked the most luxurious hotel in the world, to the Palm Jumeirah, an artificial archipelago constructed of sand dredged up from the ocean floor, the city is full of imposing sights, works of pharaonic proportions and otherworldly design.

Scrubbed clean of its history and with a majority population of foreign nationals, Dubai doesn’t have the feel of an Arab city. But it does have roots, and they’re on display in historic Bastakiya, a quaint complex of labyrinthine lanes dotted with restored merchants’ houses, art galleries, cafés and boutique hotels. The area also contains Dubai’s oldest building, the 1787 Al Fahidi Fort, and, inside, the Dubai Museum (tel 971 4 393 7151), with relics of the pearl trade for which Dubai was once renowned plus artifacts and antiquities from trading partners throughout the ages.

Snag a seat on an abra — Dubai’s open-air boat taxi — and cross Dubai Creek for a tour of the glittering Gold Souk, one of the world’s largest, in the district of Deira. Wander the maze of covered arcades and be ready to bargain for whatever catches your eye. For everything haute couture, head to the massive Dubai Mall (Sheikh Zayed Road, tel 971 4 362 7500, www.thedubaimall.com) and stroll the marble-floored corridors of Fashion Avenue, 440,000 square feet devoted to luxury retail. The mall also houses the Olympic-size Dubai Ice Rink (tel 971 4 448 5111) and, at the Dubai Aquarium and Underwater Zoo (tel 971 4 448 5200), a shark tank fit for a sheikh with “the world’s largest acrylic viewing panel.” (The piranha-feeding presentation is a perennial crowd favorite).

From there, it’s easy to find the Dubai Fountain. Just look for the 500-foot sprays shooting up from a 30-acre, manmade lake in the center of Downtown Dubai, a $20 billion development billed as “the most prestigious square kilometer on Earth.” The waterworks are choreographed to music and lights, and performances run the gamut from “Shik Shak Shok,” an Arabic dance tune, to Michael Jackson’s “Thriller” — oh, the prestige is palpable.

Not content to watch the water? Al Boom Diving (Al Wasl Road, tel 971 4 342 2993,http://al boomdiving.com), the U.A.E.’s leading dive operator, offers daily dives off the Dubai coast — mainly to shipwrecks — and a night dive once a week. If that doesn’t quench your thirst for thrills, try a 4×4 safari through the Arabian Desert. The trip, offered by Desert Safari Dubai (Oud Mehta Road, tel 971 4 357 2200, www.desertsafaridubai.com) includes sandboarding down the dunes, camel treks, a barbecue dinner and entertainments featuring a belly dancer.

When the desert heat becomes too much, hit the slopes at Ski Dubai (Interchange 4, Sheikh Zayed Road, tel 971 4 409 4000, www.skidxb.com), a small slice of Aspen built into the colossal Mall of the Emirates. In addition to five downhill runs and a Freestyle Zone full of jumps and rails, there’s the tranquil, child-friendly Snow Park complete with animal-shaped ice sculptures, a colored igloo and a gentle slope for sledding.

In the world of horseracing, the Maktoum family’s Godolphin Stables are hallowed ground. Named for one of the three founding stallions of the modern thoroughbred, their top-of-the-line Arabians are annual competitors in the Dubai World Cup, the world’s richest horse race, now held at the new Meydan Racecourse (Al Meydan Road, tel 971 4 327 0000, www.meydan.ae). Alcohol is served in the stands, but the Derby this is not; formal attire is required