Non-U.S. residents can certainly open a business in the United States. Unless they're eager to be held personally liable for the acts of the company, their options are to form an LLC or a C Corporation. However, without knowing more about your business, it's hard to say whether there would be other restrictions or regulations you need to abide by. You may have customs issues; you'll certainly have tax issues; and possibly immigration issues. Make sure you get counsel who's experienced in dealing with international transactions for your industry.