It has become a Top 40 popular success three times: for Timi Yuro (1963), Eddy Arnold (1965), and the brother-sister duo Donny and Marie Osmond (1975).
The musicians on the Arnold session were Grady Martin, Velma Smith (guitars), Henry Strzelecki (bass), Jerry Carrigan (drums), Floyd Cramer (piano), Bill Walker (vibes), Harvey Wolfe (cello), Pamela Goldsmith, Ruby Ann Story (violas), Brenton Banks, Solie Fott, Lillian Hunt, Martin Kathan, Shelly Kurland (violins), and the Anita Kerr Singers (vocal chorus).
In 2020, this version was selected by the Library of Congress for preservation in the National Recording Registry for being "culturally, historically, or aesthetically significant".
During 1978, country music singer Charly McClain covered "Make the World Go Away" for her second album, Let Me Be Your Baby.
Other popular singers who have covered the song include Engelbert Humperdinck, Jimmie Rogers and Tom Jones.
"Make the World Go Away" is the song playing on the radio of the car in Underwood's 2008 music video for her single "Just a Dream".
Jim Adkins of American rock band Jimmy Eat World covered the song on various dates of his first European solo tour.
In Italy there were two local versions: the first, with the title Resta solo come sei (Stay as just the way you are), with the Italian lyrics written by Leo Chiosso, was recorded in 1964 by Iva Zanicchi; the second, with the title Qualche cosa tra noi (Something between us), adapted and arranged by Maestro Giancarlo Chiaramello, was recorded in the late 1967 by the Japanese singer Yoko Kishi.
The song appears twice in the 2015 British gangster film Legend (based on the story of London's Kray twins); once as a 'live' performance in cabaret by Welsh singer Duffy, portraying Timi Yuro, in a nightclub scene, and then again when the original Timi Yuro single version is played over the film's closing credits.