Like its predecessor, the Italian version of My Little Pony: Friendship Is Magic, broadcast on Italia 1, has a different opening song, this time not sung by Cristina D'Avena. Compared to the very happy and upbeat English theme song, this one is slower and actually a bit melancholy. The short version can be found here and the full version here. It was used only for Season 1, and later seasons (and DVD releases of Season 1) replaced it with a translation of the English one.
Every Western Transformers series imported to Japan to date has different themes.
Prime is a notable case — for episodes 1-13, rather than having a new song written for it, as is usually the case for Transformers series in Japan, it used the existing song "Feeling" by K-Pop group Big Bang. It...doesn't fit very well with the show's generally dark and edgy tone. It also has a different ending called "I Believe in All".
For 4Kids's dub of Winx Club, they wrote "We are the Winx," which sounded nothing like the original Italian theme. Then their contract with Rainbow S.p.A. was ended, and the rights to the show were given to Nickelodeon. Nick created "Winx! You're Magic Now!", sung by Cymphonique Miller. It was set to the tune of season four's theme song, "La Magia di Winx Club," but was more upbeat and had different lyrics.
The Smurfs cartoon show uses two different theme songs from the American version for foreign markets during the introduction sequence.
For the Italian dub, several different theme songs performed by Cristina D'Avena were used throughout the show's run.
For international markets, the episodes featuring Johan and Peewit from the second and third seasons were aired separately from The Smurfs cartoon show. The episodes use the Season 2 intro sequence accompanied with its own theme song (and a modified title at the end).
In China it's also a very much beloved show, which subsequently introduced many people to Belgian comics. Many people wanted to hear what the theme song sounded like when sung in French . . . only to find out that the Mandarin one is a completely original (though still very much brilliant) piece of music, lyrics and all.
Like The Smurfs, different theme music is used for the intro sequence of The Snorks, compared to the American version.
An extended version of the specific theme song above was done in French.
For the Italian dub, Cristina D'Avena translated the extended French version for the first intro and then performed another theme, which can be seen here.
The European Spanish dub uses this theme song known as "Un Mundo Submarino".
The Latin American Spanish dub of the first Spider-Man TV series uses this opening performed by the Chilean singer Capitan Memo.
The American version of the Lady Lovely Locks opening was not a song, but rather a speech telling the viewer about the story of the show. The Korean version, however made their own theme song, with episode clips, clips from the original opening and a clip from the Kideo TV opening which features ''Rainbow Brite and the Popples, of all things!
When the British cartoon Stressed Eric was imported to America, its theme music was thrown out and replaced with a new one (albeit one which sounded similar to the British theme), as well as a different title sequence.
The French dub of The Little Rascals, titled Les Petites Canailles, has a vocal theme, sung by a chorus of children over a montage of clips from both the theatrical shorts and Hanna-Barbera's animated shorts. The lyrics are at the bottom of this page.
The French theme song for Ewoks was previously used for the made-for-TV films has nothing to do with the original themes, but it was released as a single, charted and got an award. link
How to Train Your Dragon 2 has Into a Fantasy in the Norwegian dub, performed by Hiccup's Norwegian voice-actor Alexander Rybak. Perhaps more a case of an "Additional Foreign Theme Song" though, as the original theme Where No One Goes is also kept in the dub, with this song playing after it in the end credits.