AnimEigo

Background
AnimEigo is an American entertainment company that mainly licenses and distributes anime. The company was founded in 1988 by Robert Woodhead and Roe R. Adams, III, and are one of the first North American anime licensing companies, as well as one of the only "original" ones to remain in business. The company began struggling in the early 2010s when many of their more popular licenses expired, but in 2013 the company began successfully reissuing titles on Blu-ray, through Kickstarter campaigns.

1st Logo (April 6, 1990-January 15, 1991)
Logo: On a / gradient background, after two slides of warning and copyright text dissolve in and out from top to bottom, the word "AnimEigo" does the same, dissolving in in a spiral pattern and dissolving out forward and to the bottom of the screen. All text is in optic yellow with a dark brown outline, the standard subtitle formatting at the company.

Trivia: Like most everything AnimEigo during the company's early days, this was produced using a Macintosh II.

Technique: The text dissolving.

Music/Sounds: A synth drone with futuristic synth warbling.

Availability: Extinct. Its sole known appearance was on their original releases of MADOX-01 and Riding Bean, as well as a British VHS of the latter.

Legacy: The single most obscure American anime logo ever, due to scarcity of original release copies of their first tapes.

2nd Logo (April 6, 1990-July 1996)
Logo: On a black background, the word "アニメ英語" in Japanese begins to write itself. The text then disintegrates to form the text "AnimEigo in English. While this happens, the byline "The Best Movies You'll Ever Read!™" fades in below the logo.

Variants:
 * A variant without the byline appears on the 1992 Central Park Media laserdisc of Urusei Yatsura: Beautiful Dreamer and before a sneak preview of Bubblegum Crisis at the end of the 1990 release of MADOX-01.
 * Later releases to use this logo, from as early as February 1994 to 1996, used the "Anime Your Way™" byline from the next logo.

Trivia: As with the previous logo, this was produced using a Macintosh II.

Technique: The text writing itself and disintegrating.

Music/Sounds: None.

Availability: It can be seen on most VHS tapes released by AnimEigo in the early '90s, such as AD Police, Bubblegum Crisis, Bubblegum Crash, and Urusei Yatsura. It also appears on the U.S. Manga Corps VHS of Urusei Yatsura: Beautiful Dreamer. One of the last releases to use this logo was the July 1996 English-dubbed release of Riding Bean.

3rd Logo (1994-)
SmG48jiqpM0 Logo: On a stone background, there are waterdrops appearing all over. The text "アニメ英語" and the byline "Anime Your Way" in Japanese are carved onto the stone. Many more waterdrops fall onto the stone and the Japanese text fades into "AnimEigo" and "Times New Roman".

Technique: The waterdrops and name transformation. It's all CGI.

Music/Sounds: The sounds of a rainstorm turning up, then going back to normal.

Availability: Current. A lot of the series AnimEigo licensed are now licensed by other companies, though this might appear on their currently-released Blu-rays. This was probably first used on the Urusei Yatsura: TV 1-10 Limited Edition Laserdisc boxset, though it can probably be commonly found on their DVDs of Urusei Yatsura (except for the second movie based on the series, Beautiful Dreamer, which was licensed by Central Park Media and released under their U.S. Manga Corps label), among other properties.