Among the most popular dog breeds in the United States are bulldogs and French bulldogs. In 2016, the American Kennel Club announced this week, bulldogs were the fourth most registered breed, and Frenchies came in sixth. (Labrador retrievers, as usual, topped the list.)

Bulldogs’ big eyes and large heads give them an endearing, almost babylike look that many people find irresistible. But their truncated snouts are also associated with a host of health problems that have caused growing alarm — and condemnation of breeding — among veterinarians, geneticists and animal welfare advocates. The outcry over short-nosed, or brachycephalic, dogs has been particularly pronounced in Britain, where the debate prompted the country’s Kennel Club to slightly revise its standards for bulldogs and other breeds.

Yet dogs are not the only pets that humans have bred to have increasingly flat faces. Cats, such as Persians, also have smushed visages that are the result of our matchmaking. More surprisingly, so do some rabbits.

That’s right, rabbits. Take, for example, the Netherlands dwarf, an undeniably cute and very tiny bunny that one rabbit breeding website describes as “a ball head set atop a ball body.” This and other breeds such as the Lionhead — a maned animal that looks like no wild rabbit you’ve ever seen hopping through a field — often suffer from dental problems, ear infections and overflowing tear ducts, according to three British animal welfare charities that are trying to draw attention to the medical woes of brachycephalic cats and rabbits.

The growing demand for flat-faced rabbits “is disastrous,” said Richard Saunders, the head veterinarian of the Rabbit Welfare Association and Fund, a British organization. “Rabbits’ teeth grow continuously throughout their whole lives and must line up exactly to wear down evenly. The short face means the bottom jaw is longer than the top one . . . and the teeth do not line up. Teeth soon overgrow causing chronic pain, lacerated mouths, abscesses and in many cases death.”

A spokeswoman for the American Veterinary Medical Association, Sharon Granskog, said in an email that “dwarf and round-head, flat-faced breeds seem to be a growing trend,” in the United States, but there’s no data on their prevalence. Malformations of the jaw and dental complications are common among short-nosed bunnies, she said, and brain lesions related to breeds with dome-shaped skulls are also an issue.

In this country, opposition to the breeding of baby-faced animals, even dogs, has gotten little traction. But there are signs of growing unease. The AVMA recently considered a new policy stating that pets “should not be bred” if they have “inherited conditions such as brachycephalic syndrome, some joint diseases, bone deformation . . . heart and eye conditions, or poor temperament.” An explanation of the proposed policy also mentioned “flat-faced rabbits” and their predisposition to dental disease due to skull shape.

That policy was opposed by some dog breeders’ groups, and the association ended up adopting a more anodyne policy that doesn’t mention brachycephaly or any disorder by name. Instead, it says the association supports research on inherited and genetic disorders, and encourages veterinarians to talk to breeders and pet owners about “the responsibilities involved with breeding and selecting companion animals.”

Lionhead rabbit