This article is from the Anime Music FAQ, by Ru Igarashi with numerous contributions by others.
Cowboy Bebop - Tchaikovsky (Waltz of the Flowers)?
Evangelion - Handel (Halleluja Chorus from Messiah)
Handel (Blessed are the Lamb, Amen from Messiah)
Beethoven (Symphony #9 - choral)
Pachebel (Kanon D-Dur)
Bach (Cello Suite #1 - Prelude)
Bach (Jesus, Joy of Man's Desiring)
Bach (II Air) [is that "II" for real?]
Excel Saga - L. v. Beethoven (short sections of 4th movement from 9th Symph)
FLCL - Kabalevsky (The Comedians - Gallop)
Geobreeders 2 - L. v. Beethoven (short sections of 4th movement of 9th Symph)
Giant Robo - Donizetti (aria from L'Elisir d'Amore)
Gunbuster - Mascagni (intermezzo from Cavalleria Rusticana)
Handel (Messiah)?
Dvorak (New World Symphony)
Hana Yori Dango - Mahler (which Symphony?)
Harlock Saga - Wagner (Ring of the Neibulung) [based on or actual use?]
His and Her Circumstances (KareKano) - L.v. Beethoven (Symphony #?)
Irresponsible Captain Tylor - Rossini (William Tell Overture)
Legend of the Galactic Heroes movie - Ravel (Bolero)
Love Hina - Erik Satie (several bars from Gymnopedie No. 1 in Episode 12)
Stravinski (Night on Bald Mountain)
Magnetic Rose - Puccini (aria from Madame Butterfly)
Nazca - J.S. Bach (?)
Patlabor III - Beethoven (adagio from Pathetique sonata)
Princess Nine - Schubert (Andante con moto from Piano Trio in E Flat)
Princess Tutu - Are you kidding? What composer isn't represented in this show? For a partial list, see (beware, there are potential spoilers): http://www.geocities.com/ruigarashi/FAQS/raa_music/princess_tutu_composers.txt
Rahxephon - A. Borodin (Polovtsian Dance No. 17 from "Prince Igor")
Read or Die - Beethoven (Symphony #9, 4th movment)
Sailor Moon - A. Vivaldi (4 Seasons "Summer")
- J.S. Bach (Toccata and Fugue in D minor)
- L. v. Beethoven (Symphony #5 & #9)
- F. Chopin (Sonata #2)
- Franz Lehar ("Merry Widow")
- Johann Strauss Jr. ("Blue Danube")
Utena - Berlioz (2nd movement of Symphonie Fantastique)
Violinist of Hamelin - Rossini (William Tell Overture)
- Bizet (Carmen Overture)
- Beethoven (Violin Concerto)
- much much more
common wedding song - Wagner (Lohengrinn)
 
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