  We define equivariant projective unitary stable bundles as the appropriate
twists when defining K-theory as sections of bundles with fibers the space of
Fredholm operators over a Hilbert space. We construct universal equivariant
projective unitary stable bundles for the orbit types, and we use a specific
model for these local universal spaces in order to glue them to obtain a
universal equivariant projective unitary stable bundle for discrete and proper
actions. We determine the homotopy type of the universal equivariant projective
unitary stable bundle, and we show that the isomorphism classes of equivariant
projective unitary stable bundles are classified by the third equivariant
integral cohomology group. The results contained in this paper extend and
generalize results of Atiyah-Segal.
