This video contains all four original openings and endings. The openings that are on the episodes that have been posted so are are slightly altered. The music is L'Arc~en~Ciel's "Ready Steady Go" but the video is actually for Porno Graffitti's "Melissa." So here are all of the series openings and endings. The video does all the openings in order and then all of endings in order.

The composer for Fullmetal Alchemist is Michiru Oshima. 
Openings
1.       "Melissa" by Porno Graffitti (Episodes 2-13, also used as ending for episode 1)
2.       "Ready Steady Go" by L'Arc~en~Ciel (Episodes 14-25)
3.       "Undo" by Cool Joke (Episodes 26-41)
4.       "Rewrite" by Asian Kung-Fu Generation (Episodes 42-51)
  • "Ready Steady Go" and "Rewrite" are the only openings playing on Cartoon Network's Adult Swim because of a deal with L'Arc~en~Ciel and Asian Kung-Fu Generation to promote their songs. The DVD releases from FUNimation include all openings in their original places and format.
Endings
1.       "Kesenai Tsumi" (Indelible Sin) by Nana Kitade (Episodes 2-13)
The alternate title used by some fansubbers is "Unerasable Sin", which is also the title given to it on the American release of the first soundtrack but not on the American DVD releases.
2.       "Tobira no Mukou e" ("To the Other Side of the Door") by YeLLOW Generation (Episodes 14-25)
The alternate title used by some fansubbers is "Beyond the Door"
3.       "Motherland" by Crystal Kay (Episodes 26-41)
4.       "I Will" by Sowelu (Episodes 42-50)
The movies Openings and endings were both done by L'Arc~en~Ciel. The opening is called "Killing Me" and the ending is "My Dear." Both songs, along with Opella (a.k.a Ready, Steady, GO) can be found on the L'Arc~en~Ciel album Awake; which I will put up for download later this week.
Episode 51 - This is the only Full Metal Alchemist episode that doesn't end with music. Alphonse reads: "Humankind cannot gain anything without first giving something in return. To obtain, something of equal value must be lost. That is Alchemy's First Law of Equivalent Exchange. In those days, we really believed that to be the world's one, and only, truth. But the world isn't perfect, and the law is incomplete. Equivalent Exchange does not encompass everything that goes on here, but I still chose to believe in its principle: that all things do come at a price, that there's an end and a way, that the pain we work through did have a reward, and that anyone who's determined and perseveres will get something of value in return, even if it's not what they're expecting. I don't think of Equivalent Exchange as a law of the world any more. I think of it as a promise between my brother and me. A promise that someday we'll see each other again."
Each of the theme songs were performed by artists under Sony Music Entertainment Japan's label, whose anime distribution unit, Aniplex, handled the production and music for the series.
All the ending theme songs and animation on Adult Swim are edited down for time. The DVD releases from FUNimation include the unedited full endings.
Episode 51's ending is from the english version.
In Japan, some time after the series' end, all four opening and all four ending songs were released together in a single CD titled "Fullmetal Alchemist Complete Best", including a special DVD containing the series' creditless openings and endings. An American music-only version of "Complete Best" was released recently.
The ending of episode 25 consists of a piece of very short music from the series followed by "Tobira no Mukou e" ("To the Other Side of the Door"). The end credits for the episode on DVD are presented on a black background.
 

I will post music videos for both L'Arc~en~Ciel and Asian Kung-Fu Generation, along with a link to download to L'Arc~en~Ciel Awake CD later this week and all this month. I might show some vidoes for other FMA bands as well and will probably post all the openings and endings for anyone who hasen't seen them. 

This video contains all four original openings and endings. The openings that are on the episodes that have been posted so are are slightly altered. The music is L'Arc~en~Ciel's "Ready Steady Go" but the video is actually for Porno Graffitti's "Melissa." So here are all of the series openings and endings. The video does all the openings in order and then all of endings in order.