It depends on what you mean by race. Yes race exists in the sense that human genotypes show a pattern of clustering of similarities and differences that reflect the traditional racial divisions. If the human population mated at random, there would be no genetic subgrouping because the chance of any individual carrying a specific gene variant would be evenly distributed around the world. But this is not the case. Humans do not mate randomly because of a number of factors, including geography, sociology and culture.
But, in the sense usually implied by racists of inalterable groups with clear boundaries and no overlapping of traits, race certainly does not exist. Despite the factors mentioned above, individuals of difference “races” can and do interbreed, exchanging genes.
As miko pointed out, in this sense race is the same as sub-species in other animals (also called ‘breeds” in most domestic animals). For example, because of geography and the fact that pumas can’t afford airline tickets, a California puma is more likely to breed with another California puma than a Florida puma. That’s why you have different subspecies of pumas, each with certain characteristics.
Because of dog breeders, a Labrador is more likely to breed with another Labrador; a poodle in more likely to breed with another poodle. That’s why you have dog breeds with different genotypes and phenotypes. But the two breeds can still interbreed. That’s why you have labradoodles.
All sub-species (race, breed) level distinctions are somewhat arbitrary because members of the same species can exchange genes and interbreed.
