The differential operators that arise in applications have complex spectra. They usually possess some essential spectrum and are often not Fredholm. In mathematics there are some amazing results in index theory for differential operators but usually the constraint of compact resolvents is imposed, that is the focus is on compact manifolds. Quite a lot is known for non-compact manifolds but not enough to cover many applications. I will survey some recent ideas focussing on the question of how to generalise spectral flow beyond the Fredholm situation.