All breeds of dogs have conditions that are common to their particular breed, and the bulldog is no different. Many conditions are in direct relation to the characteristics considered desirable in the breed.

Eyes

Entropion. An abnormal rolling in of the eyelid see corneal ulcers. Stitch skin so lashes don't rub.
Cherry Eye. A condition where the third eyelid is protruding. It may be stiched in place or removed as a last resort.
Cataract. A change in structure of the lens of the eye leading to cloudiness and usually to blindness.
Dry Eye. Can lead to corneal ulcers and cherry eyes if untreated.
Persistent pupillary membrane. A developmental abnormality where the membrane forming the iris does not form properly.

Skin 

Skin Acne. Same thing as in humans, affects the muzzle and lip areas.
Fold dermatitis. An inflammation of skin folds especially in dogs with loose skin. Bulldogs are especially susceptible in the face and tail areas.

Structural

Cleft lip. A condition where the two halves of the upper lip do not join together. Cleft palate and cleft lip are often seen together.
Cleft palate. A condition where the roof of the mouth is not closed and the inside of the nose opens into the mouth.
Hip dysplasia. A developmental malformation or subluxation of the hip joints. Causes movement to be painful in its most severe form.

Breathing problems in Bulldogs, chronic airway obstruction
 

Chronic Airway Obstruction (brachycephalic syndrome). The importance of this condition cannot be overemphasized. There are billions and billions of cells in the Bulldog's body and each one of these is forced to live a substandard existence, if the oxygen supply to them is impaired. There are treatments for this condition, see your vet. Chronic Airway Obstruction (C.A.O.) is very common in all short-faced (brachycephalic) breeds e.g. Bulldog, French Bulldog, Pug and occasionally seen in other breeds.
Elongated soft palate. The soft palate is abnormally long and causes breathing disorders.

Overheating in Bulldogs

The Bulldog's heat tolerance is a lot worse, than that of human's. Dogs do not have sweat glands all over their bodies. They can sweat only  through their pads and they can dissipate heat effectively through panting. Bulldogs suffering from chronic airway obstruction cannot pant freely and that is why they often die of overheating (hyperthermia). Corrective surgery performed in time can prevent a lot of faintings and unexpected deaths at summer dog shows and at home alike.

A Word About Wrinkles

Bulldog's face is certainly covered with wrinkles and that is his trademark for sure. This is a trademark that requires daily care to keep him comfortable and healthy. Mild soap (hibicleans or similar soap like you would use on your own face that is gentle) will work fine for cleaning out between his wrinkles. Get a damp wash cloth and put some soap on it and clean good around and under those wrinkles making sure not to get any in his eyes. You should dry the area good and apply either a gold bond, corn starch or similar powder to the area. If its irritated you may consider using a good cream such as Desitin cream. It may even be required for you to clean 2 times a day those wrinkles depending on how dirty or irritated they are.

Cleaning the Tail Area

This should be done in a about the same way as above with wrinkles. It is very much a part of the bulldog hygiene. When cleaning the tail area or tail pocket make sure to use a soft cloth and get in there real good and clean it, you should get out lots of hairs and keep this area clean and powdered as well. A bulldogs tail may be tightly packed against his back side and can be a real problem if not taken care of and cleaned on a daily or every other day. Remember that a severly infected tail is painful and may need to be removed in some cases, so its very good to keep this area cleaned well!

Tear Stains and Dry Nose

There are a few products on the market that are made for cleaning tear stains. There are some commercial ones you can get at your pet store, or you can ask your vet what he reccomends or may have for this. There are also home made remedies like peroxide and corn starch made into a paste, applied and then removed, followed by a desitin cream being applied for protection. You should also keep the area around your dogs eyes wiped off and cleaned and if its been dusty around your home you may want to apply a few eye drops to his eyes a few times a week so they do not get irritated. If his nose is real dry a light coat of vaseline will help soften it and protect it from cracking. Consult your vet for more info!