If you like TV, you will love Television Tunes. The website is a huge music archive of songs from known - and unknown - television series. As far as I can tell only the tunes of original American TV series are available.
All songs are listed alphabetically, just pick a letter and you will be presented with a huge list of TV shows beginning with that letter. There have to be way more than 1000 TV themes available on Television Tunes (update: according to the site, there are more than 30,000 theme songs available).
You can expect to find many old TV themes for series like The A-Team, Dallas, Wheel of Fortune or The Outer Limits but also relatively new shows like 24, Alias or Desperate Housewives.
A click on a name opens an embedded music player that will play the song automatically. Songs can be downloaded as mp3 files or emailed instead (the link only). Last but not least a list of related tunes is listed on that page. I have the feeling that at least some of the tunes have been recorded straight from TV which I base on the quality of some of the theme songs.
You can use the search to find the TV show that you are interested in, or browse the available theme songs by letter instead. Both options work equally well, and you may use either depending on the title of the show.
Update: It is interesting to note that newer TV tunes are still added to the site. I was able to find theme songs of TV shows such as Person of Interest, Big Bang Theory or How I Met your Mother on the site. They all played fine and the download button worked as well. The songs are not that large at all and the play time too is usually below the one minute mark.
The site is still great for looking up TV theme songs. You can however use YouTube instead as well to look up the songs, just head over to the video hosting site and search for the name of the show followed by theme song.
A relatively new feature is the "browse by tags" option. Instead of browsing by letter or using search, it is a third option to browse what is available. Tags like cartoon, NFL or Steven Spielberg are provided and will return all matching theme songs to you once you have clicked on the selection.
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