There are several breeds of dog that cannot bark and there are a handful of breeds that can, but don't bark often. Any dog breed can be trained not to bark, some requiring more perseverance than others; and surgical methods exist to hamper barking.
*Barkless Dog Breeds
The most well known breed of dog that truly does not bark is the Basenji. The breed is called "the barkless dog" and is a primitive African breed still used today for hunting. Basenjis are short-haired and come in a variety of colors but they all have a white chest, white feet and a white-tipped tail. They are athletic and energetic dogs.
Just because the Basenji is barkless does not mean it is quiet. According to basenji.org the dogs can growl, crow (like a rooster) and yodel a "baroo" sound.
Another (mostly) barkless breed is the New Guinea Singing Dog. The "singer" is a wild-type sub-species of the domestic dog related to the dingo. This breed has a short double coat that is reddish golden, tan or black with white points. It has a brush tail like a fox. The dog is extremely athletic with a penchant for mountain climbing and is described as moving more like a cat.
The New Guinea Singing Dog Conservation Society, which seeks to protect this breed with only 300 remaining individual animals, says this dog has a "wolf howl with overtones of whale song." Other vocalizations include yelps, woofs, screams, whines and bird-like calls.
The next group of dog breeds are those that don't bark often. This list is somewhat flexible but can include the Japanese Chin, Bouvier des Flanders, Miniature Schnauzers, Havanese and French Bulldog. Some of these breeds are small dogs, and with their quieter vocalizations, may make better pets for apartment living.
It is also possible to debark a dog through a surgical procedure. This involves the removal of vocal cord tissue creating a "whispered" bark. The procedure is controversial at best. Some believe that it is cruel since a dog can be trained not to bark. And if they do bark, there is a reason; muting them simply does not address whatever the issue is. Others feel it is a reasonable last chance fix for a dog that would otherwise be euthanized for an excessive barking habit.
Dogs are designed to communicate through barking but nature has provided some interesting variations to the norm with the "barkless" breeds.