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Myers wrote incidental music for television: for example, The Reign of Terror, a 1964 serial in the television series Doctor Who; the theme to All Gas and Gaiters; and the theme for the BBC's Question Time.
He is best known for "Cavatina" (1970), an evocative guitar piece that served as the signature theme for Michael Cimino's 1978 film The Deer Hunter, and for which Myers won the Ivor Novello Award. A somewhat different version of this work, performed by John Williams, had appeared in The Walking Stick. And yet another version had lyrics added. Cleo Laine and Iris Williams, in separate recordings as He Was Beautiful, helped to make "Cavatina" become even more popular.
During the 1980s, Myers worked frequently with director Stephen Frears. His score for Prick Up Your Ears (1987) won him a "Best Artistic Contribution" award at the Cannes Film Festival. He also scored several low budget features (Time Traveler, Blind Date, The Wind, Zero Boys) for director Nico Mastorakis, collaborating with Hans Zimmer. He won another Ivor Novello Award for his soundtrack to The Witches in 1991.