An early epoch of cosmic inflation, driven by a scalar field slowly rolling down a flat potential, provides an elegant solution to several cosmological puzzles. The notorious sensitivity of the slow roll potential to quantum gravity effects, presents both a challenge and opportunity for string theory to connect to observations. A very promising way to explain slow roll inflation is with an axionic inflaton field, whose flat potential is protected by a perturbative shift symmetry. However, the canonical models of axion inflation - "Natural Inflation" and "Axion Monodromy" - are now both in tension with observations and difficult to embed in a UV complete theory like string theory. I will discuss these challenges, and also a way to overcome them, with an inflationary mechanism - beyond slow roll - that is well-motivated from string theory and consistent with observations, including distinctive signatures to be searched for in future observations.