Dreamcast
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As part of their plan to launch the Dreamcast in Europe, SEGA created an official sponsorship with Arsenal FC (a British soccer club). The team wore Dreamcast branded shirts when they played in their home stadium, and SEGA branded shirts when they played away from home. Because of this, the club was mocked when they played against an Italian team, as 'sega' is an Italian derogatory slang term for masturbation.
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In 2001, Sega teamed up with Pace to create a Dreamcast/cable box hybrid. The box was apparently meant to switch between live TV and downloaded games (including demos). It would also have had an internal 40GB HDD to store data.
Contributed by retrolinkx
While playing a Dreamcast disc on a device that reads CD files usually produces a warning message, Seaman, Shenmue, and Skies of Arcadia each have unique warning messages on each disc, all provided by the game's voice cast. On Seaman's disc, the narrator will jokingly warn the player that attempting to play track one will infect their household appliances with viral diseases, and the Skies of Arcadia cast informs the player that they can't save the world from a CD player; all three Shenmue messages simply warn the player that attempting to play track one (which contains game data) would produce harmful results.
Contributed by game4brains
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While the console had an official adapter to allow for the video signal to be channel through a VGA cable, some games don't support the feature. It was later made possible to bypass this restriction through the use of a Code Breaker cheat disc.
Contributed by ABOhiccups
If there is a Puyo Puyo Fever save file on a VMU attached to the console, it's possible to press START on the system menu to toggle an alternative 3D menu that can have its viewpoint moved.
Contributed by supernintendo128
Inserting a Dreamcast disc into a CD player or a device that reads CD files, a warning message will play, stating "This is a Dreamcast disc and is only for use on a Dreamcast unit. Playing this disc on a hi-fi or other audio equipment can cause serious damage to its speakers. Please stop this disc now."
Contributed by Scrubben
Some Dreamcast games supported the Jump Pack, a haptic feedback device similar to Nintendo's rumble packs. It was sold separately and could be plugged into the controller. In Japan, the Jump Pack was named the "Puru Puru Pack", and in Europe, it was named the "Vibration Pack".
Contributed by Dr. Dream354
A Dreamcast light gun was never released in the USA until Mad Catz developed the Dream Blaster, which was reminiscent of the type 2 phaser found in Star trek. Sega did produce their own official light gun, but it wasn't sold in the United States, possibly because Sega didn't want its name on a gun in light of recent school shootings (the Columbine High School massacre).
Contributed by Dr. Dream354
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There was a camera for the Dreamcast called the Dreameye. It could only store 31 pictures in jpeg format, and was only released in Japan.
Contributed by Dr. Dream354
Visual Memory Units (VMU) each held 128KB. The combined capacity of all VMUs ever sold is approximately 1TB, enough to store all 730 games ever released for the Dreamcast.
Contributed by game4brains
The Dreamcast start-up sound was composed by Ryuichi Sakamoto, keyboardist and sometimes vocalist of the Japanese electronic band "Yellow Magic Orchestra".
Contributed by KnowledgeBase
Ironically, SEGA's GD-Rom technology was created as DVDs were not available for SEGA, and they wanted to create a disc that would be hard for pirates to copy and use.

"Sega intended to use the format to curb piracy common to standard compact discs and to offer increased storage capacity. It is similar to the standard CD-ROM except that the pits on the disc are packed more closely together, resulting in a higher storage capacity: around 1.2 gigabytes, which is almost double the storage capacity of a typical CD"

The Dreamcast is infamous for being the easiest console to pirate games on.
Contributed by retrolinkx
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Shoichiro Irimjiri claimed that the Dreamcast's logo is supposed to symbolize the "origin of power", as the universe is "like a vortex."

The PAL region logo was changed from red/orange to blue to avoid any legal conflict with German publishing firm Tivola, who also used a red swirl in their logo.
Contributed by retrolinkx
Sega made a lot of unreleased hardware for the Dreamcast including, a Dreamcast DVD Player (which was a rumored empty shell), a Zip Drive, a Swatch Access, and a VMU MP3 player.
Contributed by retrolinkx
There was a motion controller planned to release for the Sega Dreamcast. It would have been used for 'Air Nights' a sequel to 'Nights into Dreams' and was also rumored to have involved with the creation of the Sega's Super Monkey Ball series. The motion Sensing technology was eventually used for the maracas in Samba De Amigo.
Contributed by retrolinkx
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If you open a Dreamcast VMU, you can see "potato" printed on a chip, as a joke referencing a "potato chip".
Contributed by DidYouKnowGaming