TRAPPIST-1 planets are invaluable for the study of comparative planetary science outside our Solar System and possibly habitability. Both Time Transit Variations (TTV) of the planets and the compact, resonant architecture of the system suggest that TRAPPIST-1 planets could be endowed with various volatiles today. First, we derive from N-body simulations possible planetary evolution scenarios, and show that all the planets are likely in synchronous rotation. We then use a versatile 3-D Global Climate Model (GCM) to explore the possible climates of cool planets around cool stars, with a focus on the TRAPPIST-1 system. We look at the conditions required for cool planets to prevent possible volatile species to be lost permanently by surface condensation, irreversible burying or photochemical destruction. We also explore the resilience of the same volatiles (when in condensed phase) to a runaway greenhouse process. We find that background atmospheres made of N 2 , CO or O 2 are rather resistant to atmospheric collapse. However, even if TRAPPIST-1 planets were able to sustain a thick background atmosphere by surviving early X/UV radiation and stellar wind atmospheric erosion, it is difficult for them to accumulate significant greenhouse gases like CO 2 , CH 4 or NH 3 . CO 2 can easily condense on the permanent nightside, forming CO 2 ice glaciers that would flow toward the substellar region. A complete CO 2 ice surface cover is theoretically possible on TRAPPIST-1g and h only, but CO 2 ices should be gravitationally unstable and get buried beneath the water ice shell in geologically short timescales. Given TRAPPIST-1 planets large EUV irradiation (at least ∼ 10 3 × Titan's flux), CH 4 and NH 3 are photodissociated rapidly and are thus hard to accumulate in the atmosphere. Photochemical hazes could then sedimentate and form a surface layer of tholins that would progressively thicken over the age of the TRAPPIST-1 system. Regarding habitability, we confirm that few bars of CO 2 would suffice to warm the surface of TRAPPIST-1f and g above the melting point of water. We also show that TRAPPIST-1e is a remarkable candidate for surface habitability. If the planet is today synchronous and abundant in water, then it should always sustain surface liquid water at least in the substellar region, whatever the atmosphere considered.