(FORTUNE Magazine) – Now that mounting a tiger's head on the library wall is widely considered an act of environmental vandalism, what can the heavyweight sportsman do to flaunt his prowess? What avoids endangered species, yet maintains the proper mixture of huge expense, exotic location, and a titanic struggle concluding in the capture of imposing prey? The answer is deep-sea game fishing, which usually means going after tuna, sailfish, or marlin. The pursuit of big fish in deep water can become a grand, expensive passion. It possessed Zane Grey and Ernest Hemingway, who were among its pioneers. Today the sport has a large following among top U.S. businessmen. They support most of the game fish centers of the world, which are in the Caribbean, up and down the Pacific and Atlantic coasts, and off Hawaii, Panama, and Australia. Stephen Sloan, a New York real estate developer who holds 19 world records for game fish, has no trouble explaining his enthusiasm. He says, ''There's a pantheism involved. To see one of these animals come out of the sea and try to shake the hook is fabulous.'' While many fishermen still mount their trophies, Sloan notes: ''Unlike a big-game hunter, you don't have to kill. You can let the fish go and let it live.'' That pantheistic feeling can come dear. A day's charter in Mexico, currently about the cheapest place to fish, runs to at least $225 for a 30-foot boat, about the smallest most fishermen would find comfortable, and a more luxurious 38-footer rents for $450. Prices can go a lot higher. In the Caribbean, Sloan and two partners rent out the 43-footer they own jointly for $1,000 a day when they are not using it. Less-than-wealthy fishermen can spread the costs because most boats can accommodate up to four people. Once a fisherman is hopelessly hooked, he will probably have to own a boat. A fully rigged 38-foot model made by Topaz costs about $300,000, and a 60-footer fit for heavy seas made by Egg Harbor will cost around $850,000. Trolling slowly in a 38-footer through the deep blue water a few miles off the southern tip of Baja California, with sere hills jutting up behind the town of Cabo San Lucas, it is easy to see why people like Sloan spend fortunes fishing around the world. In the distance a whale slaps the sea with its tail. Giant rays fling themselves out of the ocean. A huge sea turtle drifts by. Then the skipper high on the bridge sees the scimitar dorsal of a marlin slitting the sea off the starboard beam. He accelerates and swings around to pass in front of the fish. The deckhand yanks a live mackerel out of the bait box, hooks it just above the mouth, and lets it out on about 40 feet of line. The maneuver draws the live bait slowly in front of the lazing marlin. The big fish moves tantalizingly toward the bait, but decides it isn't hungry. A couple of hundred yards off, another boat is luckier. A marlin weighing perhaps 150 pounds is dancing on its tail in its own spray as it tries to shake the hook. The deep sea off Cabo San Lucas, where the Pacific meets the Sea of Cortez, churns with fish. William Doner, a vice president of Caesars Palace in Las ( Vegas, says, ''You can get bigger fish in other places, but this is the finest all-round fishing in the world.'' He was so taken by the fishing at Baja that he bought a house there and runs a fleet of seven charter boats out of Cabo San Lucas. ''We get an average of one marlin a day on our boats,'' he says, ''and you can't do that anywhere else.'' Boats in Doner's Tortuga Fleet (reservations through 714-541-6797) rent for $275 a day. When W. Matt Parr, an Irish-born American expatriate, began attracting corporate executives to Baja to fish at his resorts after World War II, they could come only by yacht or private plane. Now jets from Los Angeles and other cities serve nearby San Jose del Cabo, opening the resort to the merely affluent, who stay at one of Parr's three stunning resorts, each with its own fishing fleet. Rooms on the American plan at the Cabo San Lucas, the oldest of his hotels, start at $65 a day per person, and boats rent for $265 a day. The hotel takes cash and traveler's checks but requires advance arrangements for payments by credit card. It has no telephone, so make reservations through its U.S. office (800-421-0777). Fishing has been good for decades in the Caribbean and along the east coast of North America. Sportsmen move north from Venezuela as the big fish follow the warming waters. Tournaments are popular all along the route of the fish, with prizes ranging from a trophy to $25,000 or more in cash. An old favorite of tournament anglers is the Bimini Big Game Fishing Club, which is open to the public and offers boat slips. Rooms start at $92 (800-327-4149). Fishermen can charter 31-foot boats made by Bertram for $450 a day. Serious fishermen also look to other islands in the Bahamas, such as Treasure Cay, Chub Cay, and Walker's Cay. Montauk, at the tip of Long Island, is one of the best sport-fishing bases along the East Coast. Beginning in July, the big boats speed out to the great underwater canyon 60 miles offshore where fishing for marlin and tuna is hottest. Montauk Marine Basin, operated by Carl Darenberg Jr. and Sr., charters boats of up to 60 feet for the two-day trip to the canyon and back at costs from $1,200 to $2,000. Smaller boats for fishing closer to shore cost $550 a day (516-668-5900). Coastal waters off Chile and Peru and around Hawaii and New Zealand all offer good game fishing. Tropic Star Lodge at Pinas Bay, on the Pacific Coast of Panama 150 miles from Panama City, is exclusively a fishing resort. The lodge houses up to 28 people in weekly shifts and sends them out Sunday through Friday in a fleet of 31-foot Bertrams to seek marlin, sailfish, and less renowned game. The comfortable lodge costs $2,500 for a week's room and board plus the fishing and transportation by small plane from Panama City. The owner, Conway Kittredge, suggests reservations as much as a year in advance (305-843-0124). Those who want to land a ''grander'' -- a fish of 1,000 pounds or more -- inevitably turn to Australia's Great Barrier Reef. Bill Doner, whose hallway in Cabo San Lucas is dominated by the massive head and bill of a 1,234-pound black marlin he caught off Australia, figures he spent about $15,000 for one week's fishing off the reef. Qantas flies twice a week from San Francisco to Cairns in Queensland, one base for fishing the reef. Fishing International (800-331-7600) can arrange trips to Australia. While no major game fish are endangered, some are getting scarce in places. Partly as a result, fishing customs are changing. Some anglers will no longer kill fish they do not plan to eat. Light tackle, which requires more skill in landing a big fish, is increasingly popular. Matt Parr of Baja California is especially proud of his world record for catching a 205-pound striped marlin on line rated to stand only an 8-pound pull. That may seem something like leading an angry elephant through Times Square on a string, but even a little drag tires a big fish eventually, and the clever angler can bring him alongside. Competitions in which the angler kills his fish are giving way to tag-and- release competitions. When the fish is brought alongside, the deckhand implants a numbered plastic dart in its shoulder and releases the fish to live and hit the hook another day. But for anglers whose lives cannot be compleat without a giant trophy above the mantelpiece, game fishing still provides a chance to win one without worry of imperiling a species.