When Half Moon opened in 1954, the 400-acre resort, with two miles of oceanfront, including a crescent of sand that gave it its name, described itself as creating an oasis of relaxed gentility. And the description still works pretty well. During the day, there is enough to keep even the most active families busy—golf, tennis, horseback riding, swimming with dolphins. But in the evening, when the light is fading from Sunset Beach, and kids worn out by a day of discoveries are starting to fade, too, the resort takes on a more formal and romantic air, as anyone will discover who tries to get into the Sugar Mill Restaurant wearing shorts. The approach has made it popular not only with families, who come back generation after generation, but newlyweds, couples on their second honeymoon, celebrities, and even a surprising number of British royals, the latter two groups particularly appreciative of the luxury and privacy of the resort’s 31 villas, each with its own pool.