Authors:Turbet et alAbstract:Radial velocity monitoring has found the signature of a Msini=1.3~M⊕ planet located within the Habitable Zone of Proxima Centauri, (Anglada-Escud\'e 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 to simulate Proxima b's atmosphere and water cycle for its two likely rotation modes (1:1 and 3:2 resonances) while varying the unconstrained surface water inventory and atmospheric greenhouse effect.We find that a broad range of atmospheric compositions can allow surface liquid water. On a tidally-locked planet with a surface water inventory larger than 0.6 Earth ocean, liquid water is always present, at least in the substellar region. Liquid water covers the whole planet for CO2 partial pressures ≳1~bar. For smaller water inventories, water can be trapped on the night side, forming either glaciers or lakes, depending on the amount of greenhouse gases. With a non-synchronous rotation, a minimum CO2 pressure is required to avoid falling into a completely frozen snowball state if water is abundant. If the planet is dryer, ∼0.5~bar of CO2 would suffice to prevent the trapping of any arbitrary small water inventory into polar ice caps. More generally, any low-obliquity planet within the classical habitable zone of its star should be in one of the climate regimes discussed here.We use our GCM to produce reflection/emission spectra and phase curves. We find that atmospheric characterization will be possible by direct imaging with forthcoming large telescopes thanks to an angular separation of 7λ/D at 1~μm (with the E-ELT) and a contrast of ∼10−7. The magnitude of the planet will allow for high-resolution spectroscopy and the search for molecular signatures.