  This paper extends Dolbeault cohomology and its surrounding theory to
arbitrary almost complex manifolds. We define a spectral sequence converging to
ordinary cohomology, whose first page is the Dolbeault cohomology, and develop
a harmonic theory which injects into Dolbeault cohomology. Lie-theoretic
analogues of the theory are developed which yield important calculational tools
for Lie groups and nilmanifolds. Finally, we study applications to maximally
non-integrable manifolds, including nearly K\"ahler $6$-manifolds, and show
Dolbeault cohomology can be used to prohibit the existence of nearly K\"ahler
metrics.
