(FORTUNE Magazine) – It's a little strange how some folks still think about the airline business. There are the big players, they'll tell you, like Delta, United, and American. And then you have the smaller fish. The low-cost carriers, led by that wacky Southwest Airlines, which they mention almost as an afterthought.

Now, hang on a minute. Let's look at those "industry leaders" and ask: Big like how? Well, United parent UAL filed the largest bankruptcy in aviation history ($25 billion in assets) in December 2002. That's big. American is weighted down with nearly $18 billion of debt on its books. That's pretty big. And finally, the three large airlines lost a total of some $5.8 billion last year. That's big too.

Now let's look at Southwest Airlines. Last year the company earned $442 million--more than all the other U.S. airlines combined. Its market capitalization of $11.7 billion is bigger than that of all its competitors combined, too. And last May, for the first time, Southwest boarded more domestic customers than any other airline, according to the Department of Transportation. Sure, the majors still have more revenue--Southwest, with about $6 billion in sales in 2003, ranks only No. 7 in that department--and they have more planes and carry more passengers when you include their overseas routes. And yes, some analysts question whether Southwest's amazing growth trajectory can continue. But, bottom line: Is there any question which company is the leader of this industry?

No wonder Southwest has landed in the top ten of FORTUNE's Most Admired Companies in each of the past six years--a distinction shared only by Berkshire Hathaway, General Electric, and Microsoft. Its accomplishments would be estimable in any industry. (Southwest was the nation's best-performing stock from 1972 through 2002, according to Money magazine, up a gravity-defying 26% per year.) But that Southwest has achieved this measure of success in the snakebit airline biz is nothing short of astonishing. What's more, Southwest has sustained that success--and its grip on the top ten list--even nearly three years after its eccentric founder, Herb Kelleher, stepped down as CEO (he's still chairman) and after a swarm of upstart airlines, from JetBlue to Ted, have tried to horn in on its formula.

To figure out how, you could do worse than go back to the airline's conception. Southwest famously began 33 years ago when Kelleher (a lawyer by training) and a partner drew up a business plan on a cocktail napkin. Through decades of battling the big airlines, Southwest hasn't really changed its original formula. It enters markets in which traditional airlines hold sway and then blasts them with much lower fares. Southwest flies "point to point" (city to city), ignoring the hub-and-spoke model of most other airlines. It flies only 737s. It serves no meals, only snacks (peanuts, mostly). It charges no fees to change same-fare tickets. It has no assigned seats. It has no electronic entertainment on its planes, relying instead on relentlessly fun flight attendants to amuse passengers.

That formula has so far proved unbeatable. Consider Southwest's success against just one old-line competitor: US Airways. According to analysis by Michael Roach, an industry consultant with Unisys R2A, a division of the technology company, when Southwest entered the San Francisco--Southern California markets in the late '80s, US Air had a 58% market share in those routes. By the mid-'90s, Southwest had driven US Air completely out of them. In the early '90s, Southwest entered Baltimore Washington International Airport (BWI), where US Air had a significant hub; now US Air is down to 4.9% of the traffic at BWI, while Southwest ranks No. 1 with a 47% share.

While it's hard for Southwest to play the underdog these days--and it certainly isn't sneaking up on anybody anymore--it's still the industry maverick. No matter what its competitors say or do, no company walks the talk quite like Southwest. It's iconoclastic, quirky, and sometimes just plain bizarre. Southwest has so much insouciance, in fact, that it has allowed itself to become the subject of a reality TV show on cable channel A&E called Airline. ("We all have our baggage" is the tagline.) The cameras follow Southwest employees--who don't always come across as sugar and spice--as they deal with all manner of crisis: Nasty, staggering drunks. Passengers who stink like Limburger cheese. Chicago thunderstorms. (And yes, the show has been successful enough that A&E wants to renew for next season.) Hard to imagine most admired company Wal-Mart subjecting itself to that kind of scrutiny!

If competitors are trying so hard to copy Southwest, why in the names of Orville and Wilbur Wright haven't they been able to duplicate its success? "Because they don't get it," says Southwest's idiosyncratic president and COO Colleen Barrett. "What we do is very simple, but it's not simplistic. We really do everything with passion. We scream at each other and we hug each other." There's no question that the other airlines practice the screaming part. They haven't been so good at the hugging.

Barrett, a 59-year-old Vermont-born grandmother with a long, gray ponytail, has been with the company since the beginning. In fact, she started as Herb Kelleher's legal secretary. Kelleher, 72, is one of the most unusual businessmen in our country's history. He has some of the best people skills on earth. He is also a walking paradox. As brilliant as he is batty, Kelleher is half P.T. Barnum, half Will Rogers, half Clarence Darrow, and half Jack Welch. (Yes, that adds up to two men--but if you drank as much Wild Turkey and smoked as many cigarettes as Kelleher does, you'd be seeing double too.) Despite battling prostate cancer five years ago and turning over the CEO reins to his longtime protege Jim Parker in August 2001, Kelleher is still intimately involved in the company, handling critical government affairs, scheduling, aircraft purchasing, and strategic planning. The public face of Southwest Airlines, however, now belongs to Parker and Barrett.

To truly understand why this company continues to be such a hit with customers, you have to go behind the wall and take a look. Pay a visit to Southwest's headquarters just off Love Field in Dallas, and you'll probably think you've wandered onto the set of Pee-wee's Playhouse. The walls are festooned with more than ten thousand picture frames--no exaggeration--containing photos of employees' pets, of Herb dressed like Elvis or in drag, of stewardesses in miniskirts, of Southwest planes gnawing on competitors' aircraft. Then there are the teddy bears, and jars of pickled hot peppers, and pink flamingos. There is cigarette smoking, and lots of chuckling, and nary a necktie to be seen.

"To me, it's comfortable," says Barrett, who as the ultimate keeper of the culture sits at a desk with a burning scented candle surrounded by the densest zone of bric-a-brac. "This is an open scrapbook. We aren't uptight. We celebrate everything. It's like a fraternity, a sorority, a reunion. We are having a party!" she says, throwing up her hands. I ask Barrett how much her annual picture-frame budget is. "Oh, I couldn't tell you that," she says. "Let's just say that I first gave out the framing work to this hippie fella, and now he has a business with 13 employees."

Okay, but come on. This is a serious business, right? So how does Southwest reconcile this insanity--studied though much of it is--with the fact that it flies 5.5 million people through the air each month? (Southwest--knock on Kelleher's head--has never had a fatal crash.) "Yes, our culture is almost like a religion," says company CFO Gary Kelly, "but it's a dichotomy. In many ways we are conservative. Financially, for instance." Indeed, Southwest is the only airline that maintains an investment-grade rating on its debt--a remarkable accomplishment in that business. And keeping a hawk's eye on costs is just as much a part of the company's culture as its silliness.

Yet lately, even though the airline still has some of the lowest expenses in the industry, costs have been climbing at Southwest. A key metric used in the business is cost per available seat mile (CASM), and in 1995 Southwest's CASM was 7.07 cents. Today it is up to 7.60 cents. One reason: Fuel prices have risen significantly since then. Another reason: higher compensation. "Southwest's pilots have been getting greedy," says industry consultant Roach. "They seem to think that because they work at the best airline in the business, they should get paid the most."

But let's put the cost creep in context. The big carriers all have CASMs of between 9 cents and 13 cents, and they haven't been closing the gap on Southwest. In other words, their costs are increasing at the same rate. (True, JetBlue has been able to achieve costs below Southwest's, but its have also been climbing recently--which in part explains why JetBlue's once highflying stock is off some 50% from its peak.) And Southwest management is working hard to keep a lid on costs. For instance, late last year the company announced it was closing down three call centers to save money--more than $20 million annually--as more of its customers make reservations online. "We are not satisfied with these inflationary trends," says Southwest CEO Jim Parker dryly and determinedly.

Keeping costs under control and keeping its culture alive: These are huge challenges for Southwest as it moves from upstart to prime-time player. The company now has 34,000 employees and flies to 58 cities (59 when it opens up in Philadelphia in May). Las Vegas, with 185 daily departures, has become Southwest's most-served airport. It has a fleet of almost 400 jets, with hundreds more on order (which will be painted deep "canyon blue," replacing the carrier's familiar brown and red). That's big.

So what about it, Herb Kelleher? Is the company losing its soul, as some critics have said? "No," says Kelleher, puffing on an early-morning Merit Ultra Light. (I opt for a PayDay candy bar, which he keeps in a jar on his desk--"because I drink," he says.) "It'd better not be, because I'm not going to be around forever," he laughs. "Listen, we have an incredible esprit de corps here. It's like the Marine Corps. The intangibles have always been more important than the tangibles. Plus we run this company to prepare ourselves for the bad times, which always come in the business."

Of course, the airline industry is just now emerging from the absolute worst of bad times--9/11 and its aftermath. While the rest of the industry laid off thousands of people and lost more than $22 billion over the past three years, Southwest didn't furlough a single employee and remained in the black every quarter. In fact, it has kept its string of profitable years alive at 31 straight. That's because, unlike its competitors, Southwest has wide enough margins to take a hit.

Even though Southwest has mostly flown above the storm clouds during the past few years, the world has changed dramatically for this business and this company. In JetBlue and AirTran, Southwest faces a couple of strong, innovative, low-cost competitors (see table). Orlando-based AirTran has been growing fast, operating primarily out of its Atlanta hub--yes, a hub. JetBlue and its high-profile CEO, David Neeleman (who once worked at Southwest), have managed to generate a huge amount of buzz with their cool new Airbus A320s outfitted with DirecTV at every seat (great for kids!). Though JetBlue is still much smaller than Southwest--it has some 50 aircraft and a hundred more on order--and doesn't yet directly compete with any of Southwest's routes, this newbie has definitely caught Jim Parker's attention. For instance: "We have looked and are looking at in-flight entertainment," Parker says. "Right now it just costs too much."

Delta's Song and United's Ted--the low-cost airline-within-an-airline concepts--provide Southwest with another set of challenges. Or do they? Already Song has delayed a planned expansion of its cross-continental routes. As for Ted, it's too early to tell--but its parent, UAL, which is still in Chapter 11, obviously has huge issues. (Read: very high costs.) While some see Song and Ted as parental saviors, Susan Donofrio of Deutsche Bank Securities is less sanguine. "Song and Ted aren't real, viable competitors to Southwest," she says. "They are Band-Aid fixes. The mainline carriers have to address their fundamental cost issues."

As for Kelleher, he's content merely to say, "I've seen this movie before." Perhaps left unsaid is, "And it doesn't end well." Indeed, a recent report of Donofrio's contains a list of 13 low-cost carriers that have filed for bankruptcy since 1991--and that's not counting Kiwi twice and Midway three times!

Meanwhile, Kelleher and Parker continue to do what they do best: take aim at the big guys. On May 9, Southwest will enter Philadelphia, a stronghold of beleaguered US Air, which emerged from bankruptcy last year. To Kelleher, the City of Brotherly Love holds special significance. "I grew up in New Jersey," he says, eyes ablaze. "Philadelphia was my city. I bought my first suit there. Went to my first dance there." More to the point, the Philadelphia market is overpriced and underserved, Kelleher says, a problem that Southwest is going to "cure." With a metropolitan-area population of more than seven million, "it is the nation's eighth-largest city, but its airport is only the 18th busiest," Parker points out. Overall, industry consultant Roach expects that Southwest's fares--one-way to Providence for as low as $29, to Orlando for $79--will be 25% to 75% lower than US Air's. Some say US Air's very survival is at stake. "We're not going to run away and hide," says a US Air spokesman. "We will be a vigorous competitor to Southwest in Philadelphia on every route they fly."

High-stakes jousting with the majors. Squeezing every nickel. All the while keeping the fun level cranked up to the max. That's how Southwest does business. No question, it's a tricky and singular model. And no question it all begins with Herb Kelleher. So what happens when Kelleher finally does depart from the company? Roach of Unisys puts it thus: "I never thought of Southwest as just the Herb Kelleher show. I look at it like Christianity or Islam. It was started by one guy, but it sure keeps on going." Much to the chagrin of its competitors. And much to the delight of its employees, customers, and shareholders.

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