I have a family member that breeds dogs quite seriously - one of their goals is to reintroduce good health traits back into the (working) breeds and undo the awful effects of inbreeding. Sometimes this involves reading the breeders notes from over a hundred years ago and trying to reproduce the same result (it's not easy). Often it seems to be enough to introduce a few closely related healthy breeds and after three or so generations you can get back to something healthier and a good representative of that breed.
There are a few breeds that are in desperate need of serious dedication - namely short-nose breeds (i.e. Pugs, British Bulldogs, etc). Pugs are by far the worse I've encountered in terms of physical health defects, a significant number of Pug puppies are culled early on due to spine defects of breathing issues. Those that end up getting sold are almost guaranteed to end up with breathing issues.
One of the healthier dogs you can own tends to be a mid-sized Mutt (cross-breed). From annecdata, I've found that a Bulldog (typically strong working dogs, dumb but loyal) crossed with something smarter but mid-sized tends to yield good temperament, long-life dogs with average intelligence (which is a good thing [+]).
[+] Generally you want your dog to be smart enough to read your queues and work around you, but not so smart that it's getting bored or trying to figure out how to escape.