This article is from the Anime Music FAQ, by Ru Igarashi with numerous contributions by others.
Evangelion - "Fly Me to the Moon" by Bart Howard (1954), most notable version sung by Frank Sinatra
FLCL - many songs by The Pillows [needs confirmation of prior release]
Maison Ikkoku - "Alone Again" by ?, "Get Down" by Gilbert O'Sullivan(?)
Metropolis - "I Can't Stop Loving You" by Don Gibson, sung by Ray Charles.
Serial Experiments Lain - "Duvet" by Boa. "HimuraLain" recommends checking http://www.boaweb.co.uk in the biography section to explain the circumstances around that particular acquisition by Pioneer.
Teki wa Kaizoku (Enemy's the Pirate) - "It's Only Love", "Danger on the Street" and "Big Beat, No Heart", by Lea Heart, Paul Dianno, Biff Byford. [need confirmation of prior release]
Urusei Yatsura - 'almost every piece of the Beatles' "Magical Mystery Tour" album... (Along with the occasional disco-ized "When You Wish Upon a Star"' - Derek Janseen
Black Heaven - "Cautionary Warning" by John Sykes. ""Cautionary Warning" first appeard on John's album "20th Century" (Mercury Japan PHCR-1590, released in 1997), two years before "Black Heaven" was released in Japan (1999)." - Glenn Shaw
 
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