Foreign Language Theme

The Opening Theme or Ending Theme is in a language other than the main language of the country of origin or of the Target Audience.

Note that this does not include the use of a few foreign words for flavour. Nor does it include cases where a show is dubbed into the local language but has its theme song left intact. This trope is strictly about theme tunes which are mainly or entirely in a language that the target viewers do not speak in everyday life and the original voice actors do not speak in the recording sessions. An Ominous Latin Chanting is, of course, in a language that nobody speaks anymore.

May overlap with Real Song Theme Tune.

Examples of Foreign Language Theme include:

Anime

Both openings of Ghost in the Shell: Stand Alone Complex are Russian with a little bit of Latin and English thrown in.
The Ending Theme of Paradise Kiss is Franz Ferdinand's "Do You Want To?".
Bleach's first ending, performed by Rie fu, is half Japanese, half Surprisingly Good English.
- A later opening, "Tonight Tonight" is entirely in (not particularly good) English.
So is the third ending of Gundam SEED Destiny, sung by the same person from above.
Deadman Wonderland
May I present "Human Touch", the first ED of Gundam X by the late great Warren Wiebe.
The first ending to Mobile Suit Gundam 0083: Stardust Memory is called "Magic" and sung entirely in English.
Serial Experiments Lain's OP is in English. And is English, sort of, it's by a band from London.
The Opening of .hack//SIGN is sung in English (with some unintentional Engrish as well) but it's ending is in Japanese.
Both the opening and ending of Wolf's Rain are in English.
Ditto for Gasaraki.
The ending of Neon Genesis Evangelion is a stanza of "Fly Me to the Moon". Sometimes good, sometimes not.
- Asuka's English voice actress dubs one of the versions, "Fry Me to the Moon".
Namie Amuro's "Come", one of the Inuyasha endings, is half in English, including the chorus.
Samurai Champloo's opening is entirely in English, as is "You", the end theme of one specific episode.
So is the ending of Higurashi no Naku Koro ni.
And the opening of Ballad of a Shinigami.
Gunslinger Girl‍'‍s opening is in English and performed by The Delgados.
The ending of Hellsing is in English, performed by Mr. Big.
The Japanese version of Speed Grapher used Duran Duran's "Girls on Film".
So are Anna Tsuchiya's "Rose" and "Lucy", both openings to Nana.
"Hello, Vifam," the opening theme of Ginga Hyouryuu Vifam, is in English.
Mushishi's opening theme is "The Sore Feet Song" by Scottish singer/songwriter Ally Kerr.
Gankutsuou has an English opening and ending sung by Jean Jacques Burnel, lead singer of The Stranglers.
Licensed by Royal has an opening by Billy Preston.
Ergo Proxy uses Radiohead's "Paranoid Android" as its ending theme, and while MONORAL, the artists behind the opening theme "Kiri", are Japanese, it's sung in perfect English (the band members are bilingual).
Oban Star Racers' ending is in Japanese. Its opening combines Japanese and French.
Elfen Lied OP in Latin (with a bit of Greek), ED in English, and titles in German.
Some episodes of Tsukuyomi Moon Phase have a French ending.
Black Heaven has John Sykes's "Cautionary Warning".
Black Lagoon's opening is in English... Although it may take most English speakers a few listens to realize it.
Same deal with Rideback. It and Black Lagoon's were written by the same artist.
Legend of Galactic Heroes has four different openings, all of which are entirely in Surprisingly Good English.
Naruto's first ending, "Wind", is almost Surprisingly Good English.
Opening and ending of Animal Detectives Kiruminzoo are sung in Thai, by Neko Jump.
All of the songs by the Boom Boom Satellites (a Japanese band that sings in either Surprisingly Good English or Engrish) and Kylee (American born girl of Japanese descent) for Xam'd: Lost Memories.
Monster's first ending theme is in English.
The closing theme to Gun X Sword is the full-English "A Rising Tide".
One of Black Butler‍'‍s closing themes is English "I'm Alive".
Eden of the East's opening "Falling Down" by the English band Oasis.
Princess Principal‍'‍s themes are all in English, as befits a show set in (a steampunk version of) Victorian London. The TV series themes are "The Other Side of the Wall" and "A Page of My Story", and the Crown Handler movie series' themes are "LIES & TIES" and "Nowhere Land".

Film

The credits theme to The Matrix Revolutions, "Navras", takes its ominous Sanskrit chanting lyrics from the Upanishads.

Live-Action TV

The solo at the opening of each episode of Battlestar Galactica is a Hindu prayer in Sanskrit. The expository text on screen is in English, however (and has nothing to do with the music).
Norwegian sitcom Hos Martin uses the fully English title theme "Lord of the Fools"
While it was usually in English, Weeds had at least one instance where the theme was sung in French, and has probably done another language by this point.
An episode of Psych, "Lights, Camera... Homicidio", has the theme song translated into Spanish.
- A later episode has it in Hindi.
While viewers of Monkey may not realize it, the opening theme "Monkey Magic" and the rest of the soundtrack was always Surprisingly Good English in Japan.

Video Games

The opening of Civilization IV is an arrangement The Our Father/Lord's Prayer in Swahili. It's rather catchy.
Dissidia Final Fantasy has the English song "The Messanger" by Your Favorite Enemies not as one, but three different theme tunes for the game.
The ending theme to Metal Gear Solid is in Gaelic, although sung by a Japanese singer. The Bad Ending theme to Silent Hill, by the same singer (Rika Muranaka), is in Spanish.
"Dragonborn", the theme of The Elder Scrolls V: Skyrim, is in dragon language.
Command & Conquer: Red Alert 3 had a title theme in such bombastic, overwrought Russian that no one can decipher it. Its expansion, Uprising, had a theme in Japanese instead, about its Breakout Character Yuriko.

Western Animation

One episode of Animaniacs had its opening title sequence entirely in French.
Teen Titans has both an English and a Japanese version of its Opening Theme. There is some correlation between the language of the opening and the seriousness of the episode; the English version is a basic Expository Theme Tune, whereas the Japanese one is borderline nonsense talking about, among other things, how they obey all traffic rules.
Lilo and Stitch has its soundtrack in Hawaiian and Lilo & Stitch: The Series has its theme song in Hawaiian as well. Its Japanese counterpart Stitch! has some of its theme songs in a mix of English, Hawaiian, Japanese, and Ryukyuan.
- Though the speed in which the series runs its theme song renders it Foreign Sounding Gibberish; Hawaiian isn't really compatible with the Motor Mouth effect.