A Purebred cat is one that has been bred from members of a recognized breed or strain without mixing other breeds/strains for many generations.
A purebred has been proven to breed true with respect to creating offspring of the same relative characteristics and type.
A purebred cat is an attempt of responsible breeders to conform to a set of standards set for their breed group.
A purebred cat will be recognized as such by having been registered as a purebred with a recognized registering body such as TICA, The International Cat Association.
Many persons would like to breed their cat in hopes of getting a kitten like their kitten. 
That is very unlikely to happen if your cat is not a purebred. In litters produced by non-purebreds, it is more likely that the litter will show no uniformity in size, color or any other characteristics.
| What is Considered A Responsible Breeder |