After the success of "Snowbird", Murray had a number of subsequent singles that charted both pop and country simultaneously. During the 1970s and early 1980s, her hits included Kenny Loggins' "Danny's Song" (1972) (peaked at #7 on the Hot 100) and "A Love Song" (1973); "He Thinks I Still Care" and her Top 10 cover of The Beatles' "You Won't See Me" (1974); her all-time career-peaking #1 Hot 100 hit "You Needed Me" (1978) — oddly, though, the biggest pop and commercially successful hit of her career (and, she claims, her personal favorite song in her entire repertoire) stalled out at #4 on Billboard's country singles chart and #3 on Billboard's U.S. Adult Contemporary chart; "I Just Fall In Love Again," "Shadows in the Moonlight," and "Broken Hearted Me" (all from 1979); her revival of The Monkees' 1967 #1 hit "Daydream Believer" and "Could I Have This Dance?" from the Urban Cowboy motion picture soundtrack, both from 1980; "Blessed Are the Believers" (1981); "Another Sleepless Night" (1982); "A Little Good News" (1983); 1984's "Just Another Woman in Love" and "Nobody Loves Me Like You Do" (a duet with Dave Loggins of 1974's "Please Come to Boston" fame and cousin of Murray's frequent songwriter Kenny); and "Time, Don't Run Out On Me" from 1985.

Anne Murray's Greatest Hits is the fourth greatest hits album by Anne Murray issued by Capitol Records in 1980. It is a collection of ten singles released between 1970 and 1980. "Could I Have This Dance" was newly recorded for this album. The album peaked at #16 on the Billboard 200, #2 on the Top Country Albums chart and is certified 4× Multi-Platinum by the RIAA.

1. "Snowbird" (Gene MacLellan) - 2:10
2. "Danny's Song" (Kenny Loggins) - 2:56
3. "A Love Song" (D.L. George, Loggins) - 2:46
4. "You Won't See Me" (John Lennon,Paul McCartney) - 4:08
5. "You Needed Me" (Randy Goodrum) - 3:38
6. "I Just Fall in Love Again" (Stephen Dorff, Larry Herbstritt, Harry Lloyd, Gloria Sklerov) - 2:47
7. "Shadows in the Moonlight" (Charlie Black, Rory Michael Bourke) - 3:25
8. "Broken Hearted Me" (Goodrum) - 3:43
9. "Daydream Believer" (John Stewart) - 2:26
10. "Could I Have This Dance" - 3:16
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