I'll outline the construction and computation of a Floer homology theory for contact manifolds which are prequantization spaces over monotone symplectic manifolds, and of the spectral invariants resulting therefrom, and present some applications. These include a quasi-morphism on the universal cover of the identity component of the contactomorphism group of the real projective space with the standard contact structure, the existence of a translated point for any contact form for the standard contact structure on any prequantization space over a monotone symplectic manifold (subject to a noninvertibility condition on the Euler class), and a proof that the Reeb flow of the standard contact form on such a prequantization space gives rise to a noncontractible loop in the contact group. The construction involves Lagrangian Floer theory in a nonconvex symplectic manifold. This is joint work in progress with Peter Albers and Egor Shelukhin.