I don't have the space to host full videos, so most of these are external links.  Please let me know if you see one is dead!

Court Records now has a youtube account of it's own, with all these videos and more in it's favorites.  Click here to view the trailers playlist.

GS3 Trailer
This trailer for GS3 was shown at the Tokyo Game Show, 2003.  It features several full-voiced quotes from the first two games.  The music remix played in the background is part of the GS3 OST bonus disk.
TGS 2005 (eng) This promotional trailer for Yomigaeru Gyakuten was shown at the Tokyo Game Show, 2005.  It features Maya on trial for stealing Gumshoe's lunch.  Linked is the original and an English subtitled version.
PW Trailer Trailer for PW:AA.
Justice For All Teaser This trailer for JFA was shown at E3, 2006.
Justice For All Trailer This trailer for JFA was released by Capcom shortly after E3 2006.
Trials And Tribulations Trailer The first trailer for AA3, in April 2007.
GS4 Trailer
Youtube / Direct
This official GS4 trailer was shown at the Tokyo Game Show 2006.  You can read up on more about the game in the News section (and no, that's not Edgeworth in the last shot).  You can also see the same trailer in French.
TGS 2006

Youtube / Translated part 1 / Part 2

This promotional trailer for GS4 was shown at the Tokyo Game Show, 2006.   It covers the trial of Kamen Mask the Fourth, charged with stealing GS4's script.
Tsutaya GS4 Promo
Youtube/ Direct
Another GS4 trailer showing some gameplay elements and a possible defendant of GS4-2.  Originally hosted at Tsutaya.co.jp.
GS4 Trailer 3
Direct / Youtube
The third promotional trailer released for GS4, including a sketch-like artistic style and the first look at Kyouya's animations.
Trials and Tribulations Trailer
Direct / Youtube
English trailer for AA3 recently released by Capcom (April 12, 2007).  T&T is due out this September!
Trials and Tribulations Trailer 2
Jeuxvideo / Youtube
English trailer released at E3 2007.  Shows a few new English names.
AJ Official English Trailer
The official English Apollo Justice trailer, with English names and voices, released at TGS 2007.  It's almost exactly the same as the one released in 2006.
AJ Trailer 2
Same trailer as the "GS4 trailer 3" but translated, and without voices.  But Sorchgid slapped the Japanese audio over the English video, here.  The words scrolling through the background match more or less to what they're saying, most likely official game translations of the original quotes.
Gyakuten Kenji Trailer
Trailer for Gyakuten Kenji released in April 08.  Also available subbed, or raw on the official site.
GK Trailer TGS
Youtube / Fan Translated
Trailer for Gyakuten Kenji released at the 2008 TGS.  Translated and subbed by CR forumers.
GK Cast Trailer
Trailer for Gyakuten Kenji with voice acting for Edgeworth, Kay, Gumshoe, Franziska, and Rou.
Ace Attorney Investigations
A leaked trailer for the English localization for Gyakuten Kenji.  Not sure if it's *supposed* to be out yet so the link may go dead soon.

If I've missed a trailer / video, email Croik at courtrecord @ gmail.com to let me know!