I will introduce a Fr\"olicher-type spectral sequence that is valid for all almost complex manifolds, yielding a natural Dolbeault cohomology theory for non-integrable structures and a rich theory surrounding it. As an application, we will see how Dolbeault cohomology may be used to detect the non-existence of compatible nearly K\"ahler metrics. In fact, for nearly K\"ahler manifolds, the Frölicher spectral sequence always degenerates at $E_2$, and so the second page recovers the Hodge-Verbitsky decomposition. I will end with a list of open questions that would be great to discuss with the participants during the workshop. This is joint work with Scott Wilson.