There were once many around-the-world stopping flights, both on carriers now confined to history books and those that still exist, including 
United Airlines. Some high-end charter-type services still exist, but if you think such operations have finished for scheduled flights, think again. According to Aeroroutes and reflected in OAG and on booking platforms, Hainan Airlines will begin a China-USA-Europe-China around-the-world service in mid-October, although it is not quite as it seems.