If MoneyGram money transfer, for some reason has not been received by the addressee, the sender will receive a refund. International company MoneyGram has expanded its service capabilities of online money transfers to UK residents. The benefits of money transfers by MoneyGram: there is no need to open a checking account when implementing the transfer. Until 2004, the owners of MoneyGram were Viad Corp and Travelers Express Company Inc. MoneyGram completed the rollout of simplified pricing for money transfer services in the U.S., an initiative that began last year.