  We study the phases of gravity coupled to a charged scalar and gauge field in
an asymptotically Anti-de Sitter spacetime ($AdS_4$) in the grand canonical
ensemble. For the conformally coupled scalar, an intricate phase diagram is
charted out between the four relevant solutions: global AdS, boson star,
Reissner-Nordstrom black hole and the hairy black hole. The nature of the phase
diagram undergoes qualitative changes as the charge of the scalar is changed,
which we discuss. We also discuss the new features that arise in the extremal
limit.
