Problem: My cat sheds a lot. Plain brushing just can't keep up with the hair. Is there a vacuum cleaner made for vacuuming cat hair off the cat?
Possible solution: There is a pet attachment "brush" for vacuums, I honestly can't recall whether it's a generic attachment that will fit most vacuum hoses. It's a round, smallish rubber attachment with stiff but flexible "tines" around the circumference, about thirty of them. It seems to work pretty well.
The bad news: Not all cats will take to it, due to the noise of the vacuum cleaner - but with an extended hose... who knows?