The Song Remains the Same

Basically, when an imported work is translated, theme songs and other songs are left in the original language. There's a really good reason for this; due to differing language structures, perfectly respectable Japanese lyrics are nonsensical in English. Rather than overhaul the lyrics in the song as is normally done with spoken dialogue, the song is played in the original language to avoid dialogue failure of Zero Wing proportions.

This is an aversion of Alternative Foreign Theme Song, where the theme of a foreign TV show, game or film is changed to a completely different one when aired in another country.

Named after the Led Zeppelin song. Not to be confused with the similarly-titled concert movie.

Anime
"Here he comes, here comes Speed Racer He's a demon on wheels He's a demon, and he's gonna be chasing after someone"
 * When One Piece was licensed by 4Kids!, they replaced the theme song entirely. It... wasn't well-received by fans.
 * When FUNimation licensed it, they used the original tune and translated the lyrics.
 * Oddly enough, before 4Kids! bought the rights to it, there was a promotional English version of "We Are" (with more pirate-y lyrics than the original) of unknown origins.
 * The title theme of Pokémon, also licensed by 4Kids!, got the same treatment.
 * Except for Chiisaki Mono, the ending theme of Anime Pokemon Jirachi Wishmaker, of which the first half of the song is rewritten with new English lyrics, then switches back to the Japanese version in the second verse, ultimately resulting in a Japanese-English duet. The full Japanese version is available as a bonus feature on the DVD.
 * Rurouni Kenshin: The English adaptation of the theme song uses the same music, but translated lyrics.
 * Sailor Moon: Same music, entirely new (rather than translated) lyrics, which do not match the theme of the original Japanese version.
 * But adds a kick-ass guitar solo.
 * In the 99th episode of the series, the insert song playing is left in Japanese. Then two episodes later, "Ai no Senshi" gets translated...with different lyrics.
 * Ranma Half: Usually plays it straight, but does avert it on (some of) the OVA releases.
 * And when someone sang the first song in-series it was changed to English.
 * For the first 4 or 5 seasons, they actually went to the effort of having the translated subtitles match the music syllables, meaning you could actually sing along if you so wanted to. Later seasons though just translated the lyrics straight up.
 * Suzumiya Haruhi With a Twist, in that it's played straight in the opening and closing sequences, but when Haruhi sings on stage as part of ENOZ, the dub actually bothers to have an English version of the song.
 * The Russian version did translate the opening and ending.
 * Averted in Nerima Daikon Brothers, dubbed by ADV Films, because the whole series is a musical and it wouldn't have made sense not to translate the songs, theme included.
 * The one exception is the Prime Minister's theme, which plays in the background when he first appears. The song was only included in the BGM track ADV received, so they couldn't remove the Japanese vocals.
 * FUNimation often averts this by adapting the original theme song into an English version (i.e., same tune, translated lyrics). Fruits Basket, Desert Punk, Detective Conan, Ouran High School Host Club, and One Piece all come to mind. Many of these dubbed themes are surprisingly good (though not all of them are as good as others).
 * Every single incarnation of Astro Boy. The original & 80s versions wrote new lyrics to the same tune (infact the 1960s English version was the first one to have lyrics. The first season of the Japanese version only had an instrumental, but new episodes made after the series was first exported & older ones in syndication added Japanese lyrics). Strangely, both American versions had completely different lyrics for the same tune, while the Japnese ones were the same in all versions. Both the Japanese & English versions of the 2003 series dispensed with the iconic Astroboy theme, the Japanese going the modern Anime Theme Song route with TRUE BLUE by ZONE, while the dub had some techno instrumental tune possibly for music licensing issues (the classic theme song is used as the ending theme for some of the Japanese episodes, though).
 * Record of Lodoss War: The closing theme is translated into English in the dubbed version. Scarily well-translated, at that.
 * Ouran High School Host Club: The opening and ending songs are translated into English surprisingly well.
 * The Speed Racer TV series: The opening theme was translated to English, but not very well:


 * What's interesting here is that the English opening is actually based on the version of the tune that plays over the original's closing credits.
 * The Spanish dub of Naruto has a rare exaggeration of this, in that not only they leave the original songs, but add subtitles.... in Japanese. With no Spanish subtitles.That sure is useful.
 * The latin american dub of some songs are changed but keep the same meaning as the song, and came out very good (or at least I think so) Like this Op from Cardcaptor Sakura
 * Played straight in the North American dub of Inuyasha. The Latin American dub, however, has some of the openings translated into, and sung in, laughably bad english. The first opening, for example. And that's probably the best one.
 * A dubbed one was soon released.
 * One episode of Slayers Next has Lina and Amelia casting a (completely useless) song by dressing in sailor fuku and doing a song and dance routine. They end up doing it twice, once doing it one line at a time, followed by them actually singing the entire song. In dubbed versions, the first pass is done in the dubbed language, while the second is done in the original Japanese. (The English dub even has Lina declaring "In Japanese!" beforehand.)
 * Tenchi Muyo!! and most of its spinoffs featured English dubbed theme songs.
 * The theme songs for Outlaw Star weren't translated, except for the two times the first ending theme is featured being sung by Mefina.
 * Yu-Gi-Oh!: The Japanese theme song was removed completely, replaced with mostly instrumentals with limited exclamations.
 * The English and Italian dubs of Puella Magi Madoka Magica leave the opening and ending theme in Japanese.

Films -- Animation

 * For the dub of Princess Mononoke, the lyrics of the theme song and Women Workers' Song were both translated. (Bizarrely, the latter appears in the Japanese version on the soundtrack.)
 * The Finnish dub of Happy Feet didn't have the songs dubbed but subbed. Usually the songs are also translated in Finland, but probably the company translating Happy Feet couldn't afford to organize translating the songs.
 * Usually, songs in the Disney Animated Canon are dubbed in Germay. But I'm Still Here from Treasure Planet was neither dubbed nor subbed.
 * The main criticism leveled against the Hungarian dub of The Nightmare Before Christmas is that the songs, which carry about as much significance as the dialogue, are left in English. What makes this even more bothersome is that the subtitles for the songs on the DVD don't even use the same name translations as the dubbed parts (although the subtitles for the spoken dialogue are the same), thus the movie makes no sense if you watch it on DVD. More fitting and beautifully translated subtitles were only available to go with the movie's television broadcasts.
 * The German (and most other) dub of Recess: School's Out leaves all of the songs in English.

Films -- Live Action

 * In the German version of Tim Burton's Sweeney Todd: The Demon Barber of Fleet Street, spoken dialogues are dubbed, songs are subbed. What makes this example odd is the fact that ca. 80 - 90 % of the dialogue is sung instead of spoken, and you have to wonder why they even bothered to dub the negligible rest.
 * After spirited discourse in German, Captain Jean-Luc Picard suddenly launches into "A British Tar" with a very British accent in the German dub of Star Trek: Insurrection. The abrupt change is made all the more noticeable by Commander Worf's sudden glance over at the now-singing captain.
 * In the English dub of Godzilla vs. the Smog Monster, the songs the nightclub singer sings are left in Japanese.

Pro Wrestling

 * Nikolai Volkoff would sing the "Russian National Anthem" before his matches, but it was really just nonsense words to the tune of the real anthem. This is because Nikolai, despite the character he played, was a Soviet defector and avowed anti-communist, and couldn't stomach the idea of glorifying the Soviet Union in song, even as part of his act.

Theater

 * Operas with lots of dialogue are sometimes performed with the songs in the original language but the dialogue translated. (For example, this performance of The Magic Flute.) In addition to preventing the inherent problems of translating song lyrics, this helps the audience follow the plot better and makes it easier on the singers, who would be familiar with well-known pieces in their original language rather than various very different singing translations, and who may be able to pronounce a foreign language by rote in a song but not speak the language well in spoken dialogue.

Video Games

 * The Tales (series) usually cuts the lyrics altogether in the opening songs. Vesperia averted the trope entirely by translating and writing English lyrics to Ring a Bell. It was even sung by the original artist.
 * Square Enix loves averting this. The result is often songs which possess the same melody as the Japanese version but different lyrics. Such as Simple and Clean in Kingdom Hearts, Sanctuary in Kingdom Hearts II, and the majority of The World Ends With You soundtrack.
 * Although they have been doing that less and less in recent years. For example, Final Fantasy: Dirge of Cerberus' theme songs "Longing" and "Redemption" (sung by Japanese rock-star Gackt) remained in Japanese.
 * Also, Advent Children's theme song "Calling" remained Japanese. So did Crisis Core's theme song, "Why".
 * The reason for the Kingdom Hearts songs having English translations is because singer Utada Hikaru was raised in America and fluent in English. The singer for FFXII's theme is half-Japanese, half-American and lived in Hawaii for a number of years. As for The World Ends With You, the singer lived in Japan but attended English speaking schools.
 * The songs for Final Fantasy Crystal Chronicles are intersting cases. The English titles are different ("Kaze no Ne/Sigh of the Wind" became "Morning Sky" and "Hoshizukiyo/Moonlit Starry Night" became "Moonless Starry Night." However, the lyrics to the English songs are actually very close to direct translations of the originals.
 * The Rune Factory series averts this; the theme songs are translated. It actually sounds like they got the same singer of the Japanese songs to peform the English ones, which unfortunately often makes it difficult to understand the lyrics at all.
 * The Wario Land 4 song played in Palm Tree Paradise is kept the same (and has hard to understand Japanese lyrics). The song itself is also in the sound test. Hear it here
 * Super Smash Bros. Brawl has a theme in Latin. Nobody speaks Latin as their native tongue. At all.

Western Animation

 * Some Polish translations of Looney Tunes cartoons feature an odd mixture: the entire cartoon is dubbed, but the songs are left in English, with a voiceover by a different (bland-sounding) actor reading the translated lyrics over the English text. (Perhaps the dubbing actors can't sing?)
 * Same case for the Korean dub of the Super Mario World Animated Adaption, though they left the insert songs alone.
 * Theme tune aside, the Code Lyoko franchise produced a whole CD of songs for the show's fake band, "The Subdigitals," in both French and English. There is one episode that features two of the songs on the CD, and the English lyrics are used in the English dub. A shame, really, because the English lyrics are kind of stupid.
 * Save for a very, very few Hungarian dubs of western cartoon shows (like Phineas and Ferb or SpongeBob SquarePants), the songs are always in English. Some shows offer subtitles (Family Guy, The Cleveland Show, South Park and at times The Simpsons). Usually, though, we don't get subtitles either.
 * While in Spain The Simpsons normally plays this straight, in the Clip Show of musical scenes, the songs were all dubbed despite originally airing in English.
 * In some foreign dubs of Recess, any scene where a group of characters are singing something is usually left in English.
 * In the Foster's Home for Imaginary Friends episode, "The Sweet Stench of Sucess", Bloo's song, "My Evil Producer Kidnapped Me and Won't Let Me Go" is kept in English in most dubs. "I'm Just Another Used-Up Deoderant Stick" is dubbed, however.