  Given an action of a reductive group on a normal variety, we construct all
invariant open subsets admitting a good quotient with a quasiprojective or a
divisorial quotient space. Our approach extends known constructions like
Mumford's Geometric Invariant Theory. We obtain several new Hilbert-Mumford
type theorems, and we extend a projectivity criterion of Bialynicki-Birula and
Swiecicka for varieties with semisimple group action from the smooth to the
singular case.
