Most people are not familiar with the term "Brachycephalic," but if you own a breed with "pushed in" faces, you should become familiar with this word. The word comes from Greek roots "Brachy," meaning short and "cephalic," meaning head.
French Bulldogs and Boston Terriers are breeds with brachycephalic syndrome. their famous flat faces are very short compared to the length of the cranium, This carries a number of health implications, some of which are quite serious, sometimes deadly. Others can negatively impact a dog's quality of life.

First, the facial bones and tissues of a brachycephalic dog are so compressed that its airway is often impaired by several related defects (stenotic nares, elongated soft palate, excess tissue in the throat, too crowded nasal cavities; and secondary changes in the larynx that these primary defects can produce over time). These are collectively called the Brachycephalic Syndrome.

The only visible part of the airway are the openings to the outside, the nares. These should be open rather than pinched, and their openings should not have been surgically enlarged. Though you can't see a dog's palate, throat, nasal cavities, or larynx, you can usually tell whether it has airway problems by watching and listening to it.

Having a big head and a lot nasal bones compacted, brachycephalic dogs tend to have trouble with the way their eyes seat in their heads. Their boney eye sockets are very shallow and don't take much for a minor injury cause an eye to pop from its socket. This can happen also with too much pulling against the leash if the pet is wearing a collar, you may wish to consider a harness for your pet.
Their eyes are so prominent that very often the lids cannot close all the way over their eyes. This could lead to irritation and drying of the center of the eye unless surgical correction is performed.
On both breeds, their eyes openings should be round, with no white showing when the dog is looking forward, and located on the front of the skull rather than on the sides (where they are situated on long-faced breeds). Boston Terriers and French Bulldogs occasionally have cherry eye (enlarged and prolapsed gland of the nictitating membrane), entropion, a visible haw (nictitating membrane), dermoid cysts (rapidly enlarging growths usually found on the margin of an eyelid or on the cornea), and juvenile cataracts. These conditions are not considered desirable, and may be inheritable.

The coat should be smooth, and shiny, and the skin soft and loose, forming wrinkles at the head and throat. However, many French Bulldogs and Boston Terriers have one or more skin problems.
The deep wrinkles of their faces should be kept clean and dry, as should the areas between the toes, to prevent infections from developing in these dark, moist areas. Good grooming, proper nutrition, good veterinary care, and an attempt to avoid breeding animals with known immune-mediated disorders are necessary to minimize the skin disorders.

The Boston Terrier, just like the French Bulldogs are also classified as a brachycephalic and chondrodystrophic breed, that results in a shortening of the vertebrae and of the long bones of the limbs. The chondrodystrophic and brachycephalic skeleton is, though characteristic of the breed, structurally abnormal, with the potential to cause some inherent physical problems.

The spine of a chondrodystrophic breed is also shortened by its abnormal type of development. Although the breed standard calls for a short and compact body, it should not be too short as the standard also calls for good proportion. French Bulldogs have a high incidence of vertebral malformations, and also of premature degeneration of the intervertebral discs. The shorter the back, the more extreme the degree of malformation of the vertebrae. As the spine is excessively shortened the size of the chest cavity is reduced, which restricts the lung capacity and compromises an already marginal respiratory system.

Excessive shortening can also affect gait, particularly if the dog is so close coupled that its gait is crabbed as it tries to prevent its hind feet overtaking its front feet. Though a Frenchie's movement is not weighted as heavily in the standard as that of many breeds, its movement should be "unrestrained, free and vigorous." If the spine is so short that there is not enough length of neck, the reach of the forelimbs will be reduced, as the neck muscles that move the forelimbs forward will be unable to shorten sufficiently to produce a good forward motion at the shoulder.

Adapted from:
MarVista Med. Center, French Bulldog club of America