You would be surprised with the types of pets that people have around the country.  Wolves, porcupines, primates...

Each state has its own laws specifying which pets are considered acceptable and which are not.  Within the state, individual cities have the right to put additional bans and restrictions on animals as they see fit.  For example, although the State of Utah considers European ferrets legal, the city of Taylorsville does not.  

Before you buy an exotic pet, especially if you do so from a private seller or abroad, be sure to familiarize yourself with Utah's laws.  Doing so may protect you from a large fine, legal problems and even jail time.  

Prohibited animals as pets in Utah

• Free-range/Game
  Bison, bighorn sheep, elk, mule deer, moose, pronghorn antelope, Rocky Mountain goat
• Small animals
  Albert's squirrel, prairie dog, Merriam kangaroo rat, desert rat, ringtail, cottontail and snowshoe rabbits, weasel, skunk, marten, mink, black-footed ferret.
• Reptiles
  Banded gila monster, desert iguana, Glen Canyon chuckwalla, Western chuckwalla, desert glossy snake, Mojave desert sidewinder, Mojave rattlesnake, Sonoran lifre snake, speckled rattlesnake, Utah milk snake, Utah mountain kingsnake, desert tortoise.
• Cats
  Tiger, lion, cougar, jaguar, leopard, cheetah, ocelot, lynx, serval.  Basically all species other than the domestic cat.
• Dogs
  Coyote, wolf, dingo, fox, jackal, African wild dog.  All Canids (Canidae) are prohibited as pets.
• Other
  Bears, wolverine, bats, river otter, badger, primates 
• Birds
  No known bird restrictions exist in Utah.  
• Fish
  Piranha, venomous fish (Orem)
   

*All species and sub-species of the mammals above that are not listed are classified as controlled. 
*All species and sub-species of the reptiles above that are not listed are classified as non-controlled. 
*The above laws do not apply to birds and fish.  Separate rules apply.

For example, a porcupine is not listed as a prohibited mammal.  Because it is not listed and is not a sub-species of any of the mammals listed, it is a non-controlled pet.  

Wolf-hybrid dogs are subspecies of wolves and are not listed specifically as a prohibited animal.  Unless the city has a law against it, wolf-hybrid dogs are controlled pets, with the exception that other than standard dog licensing, you don't need special permission to have one in Utah.

There seems to be some confusion about the legality of hedgehogs.  Hedgehogs are not listed in Utah's prohibited pet code so one would assume that they are legal. If hedgehogs are being sold in your local pet store, it is doubtful you will have a problem.  To be safe, contact the Department of Wildlife or your city government office. 

*In RARE circumstances, a person (usually not an individual) can obtain a license and certificate of registration for animals listed as prohibited.