  Because the inflationary mechanism is extremely sensitive to UV-physics, the
construction of theoretically robust models of inflation provides a unique
window on Planck-scale physics. We review efforts to use an axion with a shift
symmetry to ensure a prolonged slow-roll background evolution. The symmetry
dictates which operators are allowed, and these in turn determine the
observational predictions of this class of models, which include observable
gravitational waves (potentially chiral), oscillations in all primordial
correlators, specific deviations from scale invariance and Gaussianity and
primordial black holes. We discuss the constraints on this class of models in
light of the recent Planck results and comment on future perspectives. The
shift symmetry is very useful in models of large-field inflation, which
typically have monomial potentials, but it cannot explain why two or more terms
in the potential are fine-tuned against each other, as needed for typical
models of small-field inflation. Therefore some additional symmetries or
fine-tuning will be needed if forthcoming experiments will constrain the
tensor-to-scalar ratio to be r < 0.01.
