American Shorthair cats are generally healthy and hardy animals. However, like all breeds, they can be prone to certain health issues. Some common health issues to be aware of in American Shorthair cats include:

1. Hypertrophic Cardiomyopathy (HCM): This is a type of heart disease that is more common in American Shorthair cats. It is caused by an abnormal thickening of the heart muscle, which can lead to heart failure.
2. Dental problems: Dental problems, such as tooth decay and gum disease, are common in cats and can be more prevalent in American Shorthairs due to their small jaws and crowded teeth.
3. Obesity: Like many domesticated cats, American Shorthairs can be prone to obesity if they do not get enough exercise and are fed an unhealthy diet.
4. Polycystic kidney disease (PKD): PKD is a genetic condition that causes the formation of cysts on the kidneys. It is more common in American Shorthairs than in some other breeds and can lead to kidney failure.
5. Feline lower urinary tract disease (FLUTD): This is a common health issue in cats that can cause bladder inflammation, urinary tract blockages, and other problems. American Shorthairs may be more prone to FLUTD than some other breeds.

It is important to keep your American Shorthair up to date on all necessary vaccinations and to have regular check-ups with your veterinarian to help prevent or detect any health problems early on.