(FORTUNE Magazine) – Raymond Floyd fell in love with Shinnecock Hills Golf Club the moment he saw it. It was 1986, the Monday before the U.S. Open--America's national golf championship--and Floyd had arrived in Southampton, N.Y., for his first practice round. He had some reservations about the way the United States Golf Association (USGA) had set up courses for previous Opens, specifically its practice of growing high greenside rough at layouts from Oakmont to Pebble Beach, which he thought put a premium on drawing lucky lies rather than on shotmaking and short-game skills. But after his first practice round at Shinnecock, he told his wife, Maria, "This is the greatest golf course I have ever seen. They can't mess it up."

On June 17 through 20, the U.S. Open returns to this classic venue, a rugged stretch of turf 90 miles east of New York City and light-years from the city's noise and bustle. It was here, in 1896, that James Foulis won the second Open ever played. It was here, 90 years later, that Floyd won the 1986 Open by two shots. Nine years later Corey Pavin smacked his famous four-wood to the final green and edged out Greg Norman to win the 1995 Open. And now the likes of Tiger Woods and Phil Mickelson will battle on one of the trickiest, worthiest courses in the world.

Floyd, who went on to buy a summer home in Southampton and become a Shinnecock member, is hardly alone in his passion for the place. Shinnecock ranks No. 4 among Golf Magazine's Top 100 Courses in the World and is the highest-ranking Open host. (No. 1 Pine Valley and No. 2 Cypress Point are ultraprivate and don't host U.S. Opens; No. 3 Muirfield is in Britain.) What makes it so extraordinary? Start with the variations in the wind and weather, the elegance of the routing, the demands on driving accuracy, the emphasis on shotmaking strategy and short game. "Imagine the Masters, U.S. Open, and British Open all rolled into one," says pro Ben Crenshaw, "and you have some idea of the atmosphere at Shinnecock."

Then there's the club's history and tradition. Founded in 1891 by publishing magnate Edward S. Mead and a handful of fellow aristocrats, Shinnecock built the first 18-hole golf course on the East Coast, using mostly the labor of the local Shinnecock Indians. It constructed America's first golf clubhouse in 1892, a gracious wood-shingled building designed by Stanford White. And it hosted the first U.S. Open (in 1896) to allow African-American and Native American golfers to compete alongside whites.

Though Shinnecock lies a few miles north of the Atlantic, its rambling fairways and honey-colored native roughs conjure the look, feel, and spirit of classic seaside links. British pro-turned-broadcaster Peter Oosterhuis calls Shinnecock's landscape "more England than England." Says golf course architect Rees Jones, who has redesigned several Open courses and built the Atlantic Club nearby: "Shinnecock is unbelievably natural. The course fits the landscape like a glove." Aussie David Graham, who won the 1981 U.S. Open at Merion Golf Club in Ardmore, Pa., calls Shinnecock simply "perfect."

Much of the credit belongs to course architect William Flynn, who in 1931 completed most of the present layout. Anticipating later improvements in equipment, Flynn left room to extend the tees on par 4s and par 5s. More brilliantly, he devised a routing in which no more than two holes in a row play in the same direction. In contrast to many courses, where holes run parallel upwind and downwind, the holes at Shinnecock run crosswind as well. "No matter which way the wind blows, you're guaranteed to face a different challenge on every hole," says Floyd. "That's the genius of it."

Wind and weather are the most constant obstacles at Shinnecock, and the least predictable. The course was designed for a prevailing southwest wind, which blows at players' backs on most of the long holes. But on eastern Long Island in the summer months, the so-called prevailing wind prevails little more than half the time. In the opening round in 1986 a howling north wind accompanied by rain ballooned the average score to 78.1. (Par is 70.) "I have never seen any place, with the possible exception of Pebble Beach, where the raw elements can be as great a factor as at Shinnecock," Crenshaw says. While Shinnecock is not the longest Open venue, it may be the most challenging. (The course is 6,996 yards long, but the USGA has added about 70 yards and repositioned six tee boxes for this year's event.)

Who will win the 2004 Open? The subtle, deceptive contours of Shinnecock's greens tend to favor players with a strong short game, such as defending champion Jim Furyk. But Furyk may still be nursing a wrist injury; it's unclear whether he will be able to compete. Mickelson will be a crowd favorite, coming to Shinnecock full of confidence after his first major win, at the Masters in April. Cynics say that recent boosts in power on the PGA Tour will make "Shinny" vulnerable to just about anyone. Floyd, 61, who as a previous Open winner received a special exemption to compete this year, disagrees. He reminds us, "I'm still the only one who has broken par in a U.S. Open here."