Radial velocity monitoring has found the signature of a M sin i = 1.3 M ⊕ planet located within the Habitable Zone (HZ) of Proxima Centauri (Anglada-Escudé et al. 2016). Despite a hotter past and an active host star the planet Proxima b could have retained enough volatiles to sustain surface habitability (Ribas et al. 2016). Here we use a 3D Global Climate Model (GCM) to simulate Proxima b's atmosphere and water cycle for its two likely rotation modes (1:1 and 3:2 spin-orbit resonances) while varying the unconstrained surface water inventory and atmospheric greenhouse effect. Any low-obliquity low-eccentricity planet within the HZ of its star should be in one of the climate regimes discussed here. We find that a broad range of atmospheric compositions allow surface liquid water. On a tidally-locked planet with sufficient surface water inventory, liquid water is always present, at least in the substellar region. With a non-synchronous rotation, this requires a minimum greenhouse warming (∼10 mbar of CO 2 and 1 bar of N 2 ). If the planet is dryer, ∼0.5 bar/1.5 bars of CO 2 (respectively for asynchronous/synchronous rotation) suffice to prevent the trapping of any arbitrary small water inventory into polar/nightside ice caps. We produce reflection/emission spectra and phase curves for the simulated climates. We find that atmospheric characterization will be possible by direct imaging with forthcoming large telescopes. The angular separation of 7λ/D at 1 µm (with the E-ELT) and a contrast of ∼10 −7 will enable high-resolution spectroscopy and the search for molecular signatures, including H 2 O, O 2 , and CO 2 . The observation of thermal phase curves can be attempted with JWST, thanks to a contrast of 2 × 10 −5 at 10 µm. Proxima b will also be an exceptional target for future IR interferometers. Within a decade it will be possible to image Proxima b and possibly determine whether this exoplanet's surface is habitable.