These I feel ought to be done on every adult dog before being bred:
A very careful and thorough physical exam by vet
Complete Blood Panel
Bile Acid Test
Brucellosis Test (every time before breeding)
And these are the Health certifications available. It's very impressive to me when a breeder goes further than the basic health testing I mentioned above and gets their dogs certified for any of these things. If I ever decide to show and breed, I'd probably look into most of these for my dogs.
Patella and hips cleared & certified by OFA
OFA: Patellar Luxation
OFA: Patellar Luxation
OFA: Hip Dysplasia
OFA: Hip Dysplasia
OFA: Legg-Calve-Perthes
OFA: Legg-Calve-Perthes
Eyes checked & cleared & certified by CERF
CERF - Canine Eye Registration Foundation
CERF - Canine Eye Registration Foundation
Heart checked & cleared & certified by OFA
OFA: Cardiac Congenital Disease
OFA: Cardiac Congenital Disease
Thyroid checked & cleared & certified by OFA
OFA: Thyroid
OFA: Thyroid
Ears checked & cleared & certified by OFA
OFA: Deafness
OFA: Deafness
(personally, I think this probably isn't needed for our breed..)
You also want to do research into the pedigrees of the male and female to check for any genetic defects/diseases to identify if the dogs could be a carrier or not.