I will discuss the phenomenology and observational status of monodromy inflation. This apparently simple model is based on a stringy axion associated with a weakly broken shift-symmetry. I will show that the post-inflationary dynamics may naturally contain a phase where the universe is dominated by oscillons, pseudo-stable solutions of a nonlinear scalar field theories. Conversely, the inflationary perturbations can include a superimposed “modulation”, and I will discuss current constraints on the amplitude of these modulations, derived from the WMAP and Planck datasets.