Their Greatest Hits (1971–1975) comprises nine best-selling singles released between 1972 and 1975, as well as Desperado, which had never been previously released as an A-side. All single tracks but Tequila Sunrise charted in the Top 40, with five in the Top Ten, and One of These Nights and Best of My Love both topping the Singles Chart. 

According to the RIAA website, on February 24, 1976, the album achieved the distinction of being the first to receive the RIAA Platinum award, which was introduced in 1976, in recognition of one million shipments in the United States. In August 1990 it was certified 12× Platinum, and on November 10, 1999, it became the all-time best-selling album in the United States when it was certified 26× Multi Platinum. It was certified at 29× Multi Platinum on 30 January 2006, and has sold over 42 million copies worldwide to date. Michael Jackson's Thriller is the only other album certified 29× Multi Platinum.