The demand for shark fins is growing, as Asian countries become more economically developed. As I understand it, serving shark fin soup is essentially a display of wealth in many Asian cultures, and as more people in that part of the world become wealthier, they want to indulge in a little status symbol, kinda like buying a luxury car when you've finally gotten that great raise or new job.
Up to 73 million sharks are killed every year to support the international shark fin market, threatening already overexploited shark populations around the world. And millions more are accidentally caught as bycatch. The US actually allows finning, but the weight of fins harvested must be no more than 5% of the total shark catch (in other words, they have to bring the bodies in as well). At-sea dumping is illegal. Plus, once a shark's fins are removed, it's difficult to ID it, as fins are used to tell shark species apart. And of course, certain species of sharks are more threatened than others.
Sharks are slow-growing and long-lived animals, and often their populations cannot bounce back from the incredible fishing pressure placed upon them for their fins. In fact, one-fifth of all shark species are considered threatened with extinction according to the World Conservation Unions (IUCN) 2006 Red List of Threatened Species.