Inherited Problems in Bulldogs:

www.bulldoginformation.com  is a great resource for more information on the following:
 

Cherry Eye: A swelling of a gland in the inner eyelid. This is a common condition in bulldogs and is usually treated by cutting the gland out if the swelling is a reoccurring problem. 

Elongated Soft Palate: This occurs in bulldogs and other short-faced breeds. It often results in some degree of obstruction of the dog's airway. This obstruction results in snorting, snoring and breathing through the  mouth. In severe cases of elongated palate,the  palate partially blocks the opening into the voice box. If secondary changes in the voice box take place, acute airway obstruction may occur. If your bulldog begins to honk like a goose, put his head back, and gasp for air, consult your veterinarian to see if he or she considers your bulldog a candidate for the surgery necessary to correct an elongated soft palate.  

Luxating Patella: A dislocation of the patella, the small, flat, movable bone at the front of the knee. In mild case the patella, which is held in by ligaments, pops out of the groove in the femur in which it normally resides, then pops back in of its own accord. In severe cases the patella cannot return to its correct position on its own, and when it is manipulated into place, does not remain there long. A bulldog with luxating patella favors his affected leg when he walks, and when he runs, he lifts it, setting it down only every few steps. The tendency toward luxating  patella is inherited, but excess weight can aggravate that tendency. Luxating patella can be corrected by surgery.