What does an umbrella policy cover? Only the associated property?

[Reese W.]

Every article I read about an umbrella policy says the same thing...it covers lawsuits that go above normal liability limits. But what's never clear to me is, can an umbrella policy on my property be used to cover ANY lawsuit against my name? or only something that happens on that property itself? Do I need to get an umbrella policy on every property? I can't get one on one of my properties because I don't have insurance there (HOA has their own condominium policy) and it seems that they're only offered as an add-on.