What are the different breeds of rats?

Answers:
If you are referring to wild rats, they are species, not breeds. There are over 700 known species of rats according to the Smithsonian Institute. Use to have the link on this page of my website, http://spazrats.tripod.com/wildcousins.h... . Link now broken and I can't find it.
http://en.wikipedia.org/wiki/rats#specie...
The transition from Wild to Domestic
http://www.rathelp.org/wildvsdomesticate...
http://www.curiosityrats.com/infohistory...
If you are referring to pet rats......
Again, no such thing as breeds of fancy rats at this time, but breeders in the Fancy are working towards this goal. The species of rat that is typically kept as a pet is Rattus Norvegicus, a decendant of the wild Norway or Brown Rat by the same scientific name.
http://www.ratbehavior.org/ratspecies.ht...
http://en.wikipedia.org/wiki/brown_rat...
http://en.wikipedia.org/wiki/fancy_rat...
Right now rats are determined by type, or variety. There are specific varieties that are show quality and many more that are found in pet-quality rats. There are many mis-marked varieties, meaning their markings deviate from the show-quality, yet they are striking non the less.
http://search.yahoo.com/search?search=va...
Body type:
Common or Standard
Dwarf
http://www.skyclyde.com/dwarfrats.html...
Tail-less
http://www.skyclyde.com/taillessline.htm...
Dumbo-eared
http://members.aol.com/ristorat/dumbo.ht...
http://img.photobucket.com/albums/v693/i...
Standard-eared, also known as Top-eared
http://img.photobucket.com/albums/v693/i...
Coat type:
Standard or Smooth
Satin
True Hairless
Peach Fuzz Hairless
Patchwork Hairless
http://www.skyclyde.com/oddfellowshairle...
Rex
Double Rex
Velveteen
Harley
http://www.skyclyde.com/harleyii.html...
There's more
http://www.skyclyde.com/geneticscurlycoa...
http://www.lilratscal.com/variety.htm...
Color type:
Platinum
Sky Blue
American Blue
Powder Blue
Russian Blue
Fawn
Beige
Champagne
Siamese
Black-eyed Siamese
Himilayan
Chocolate
Havana
Black
Charcoal
Pearl Merle
Agouti
Blue Agouti
Pink-eyed White
Red-eyed White
Black-eyed White
And so many more colors
http://www.spoiledratten.com/breedingrat...
http://www.skyclyde.com/ukrats.html...
Marking type:
Self
Hooded
Bare-back
Capped
High-White
Dalmation
Verigated
Berkshire
Variberk
Irish
English Irish
Carnigan
Blazed
Possom-faced
Jester
Down Under
Snowflake
And there's more
Eye-type:
Dark eyes
Ruby eyes
Pink eyes
Red eyes
Odd-eyed
spazrats
"my life has gone to the rats"

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Bald
Hooded
Dumbo
Solid

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By Karen Yang

Agouti - A fancy word for the wild-type rat color. A pretty combination of warm cinnamony browns with black guard hairs.
BEW - Stands for Black-Eyed White.
Berkshire - A self rat with a white tummy and legs.
Capped - White with what looks like a wig of a different color.
Dalmatian - Like the dog! White with lots of spots.
Dumbo - A type named after the elephant. A rat with ears lower down on its head, giving it a clownish appearance.
Hairless - What it says! They are usually born with fur, but lose it as they grow up. Some are not completely hairless and have neat fuzzy patches. Since they don't have fur, they are more sensitive to cold temperatures.
Himalayan - Like the cat! Creamy white or off-white darkening to brownish on nose, rump and legs. Usually with pink eyes.
Hooded - A common color pattern. White with a different-colored hood and back stripe.
Husky - A new color pattern. Their coats look just like Husky dogs!
Irish - Similar to Berkshire, but with just a triangle of white on the chest.
Odd-Eye - A rat with one pink or ruby eye and one black one.
PEW - Pink-Eyed White. An albino-type rat.
Rattus norvegicus - The Norway rat or brown rat. The majority of pet rats are of this species.
Rattus rattus - The roof rat or black rat. Rattus norvegicus' smaller, more agile cousin.
Rex - A rat with curly fur and whiskers!
Satin - Having softer fur than most rats. Unstandardized.
Self - Any solid color with no other markings.
Siamese - Also like the cat! Blue-Point Siamese is a light blue with darker slate blue patches on nose, rump and legs. Seal-Point is cream or beige with darker brown nose, rump and legs. Both can have either pink or ruby eyes.
Silvered - Any solid color with an even mixture of silver or white hairs throughout the fur.
Tailless - Again, obvious. Difficult to breed for. Serious birth defects can result from the tailless gene. Also, since the tail helps with temperature regulation, they are much more susceptible to heatstroke.
Variegated - Usually means a hooded rat with splotces of color on its sides. Can mean any spotted pattern, including Dalmatian.

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Standard: With short, smooth, glossy fur.
Rex: With curly fur and curly whiskers.
Tailless: Complete absence of a tail, similar to Manx cats.
Hairless: Complete absence of fur.
Satin: Thinner, longer coat, with lustrous sheen.
Dumbo-Eared: Ears large and set low on sides of head.

http://www.altpet.net/rodents/rats/rf03.

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At present, all rats are shown in six Varieties:
STANDARD 鈥?With short, smooth, glossy hair.
REX 鈥?With curly hair and curly whiskers.
TAILLESS 鈥?Complete absence of a tail, similiar to the Manx cats.
HAIRLESS 鈥?Complete absence of hair.
SATIN 鈥?Thinner, longer coat, with a lustrous sheen.
DUMBO 鈥?Larger ears set on the side of the head.
Each of these six varieties is grouped into six Sections by color and body markings. There are 33 distinct colors recognized among these Sections.
Here is a site that will explain the differences in the rats.