  We construct stacks which algebraize Mazur's formal deformation rings of
local Galois representations. More precisely, we construct Noetherian formal
algebraic stacks over Spf Zp which parameterize \'etale (phi,Gamma)-modules;
the formal completions of these stacks at points in their special fibres
recover the universal deformation rings of local Galois representations. We use
these stacks to show that all mod p representations of the absolute Galois
group of a p-adic local field lift to characteristic zero, and indeed admit
crystalline lifts. We also discuss the relationship between the geometry of our
stacks and the Breuil-M\'ezard conjecture.
