If MoneyGram money transfer, for some reason has not been received by the addressee, the sender will receive a refund. Since going public two and a half years ago, MoneyGram has made great strides by pouring energy and resources into several critically important areas for business. The company continues to automate and enhance MoneyGram Today the company MoneyGram International - an independent U.S. company, whose shares are traded on the New York Stock Exchange and MoneyGram money transfer is considered one of the largest international money transfer systems without opening a bank account.